[
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU001",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU001-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in G2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU002",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU002-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in G2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU003",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU003-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in H1\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU004",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU004-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in B4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU005",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU005-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in A2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU006",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU006-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in A2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU007",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU007-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in A2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU008",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU008-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in F2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU009",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU009-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in F2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU010",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU010-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in F3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU011",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU011-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in H6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU012",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU012-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in G5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU013",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU013-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in C3 or G7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU014",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU014-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in C2 or G6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU015",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU015-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in B5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU016",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU016-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in F6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU017",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU017-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in E3 or E7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU018",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU018-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in E8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU019",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU019-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in E2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU020",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU020-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in B3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU021",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU021-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in H3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU022",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU022-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in H3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU023",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU023-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in C3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU024",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU024-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in B7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU025",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU025-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in H1\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU026",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU026-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in N12\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU027",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU027-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in F7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU028",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU028-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in E7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU029",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU029-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in H9\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU030",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU030-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in H8 or L12\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU031",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU031-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in K8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU032",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU032-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in I9 or M5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU033",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU033-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in M13\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU034",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU034-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in H12 or L8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU035",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU035-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in M7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU036",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU036-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in I8 or I12\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU037",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU037-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in I7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU038",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU038-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in F9 or J13\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU039",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black-and-white Gomoku piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU039-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black-and-white Gomoku piece should be placed in E8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU040",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU040-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in G9 or K9\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU041",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU041-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in H8 or L12\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU042",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU042-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in H8 or L12\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU043",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU043-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in G7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU044",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU044-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in A2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU045",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU045-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in G8 or K12\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU046",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU046-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in K6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU047",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU047-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in J7 or F11\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU048",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU048-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in E4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU049",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU049-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in L13\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU050",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black-and-white Gomoku pieces. The black-and-white Gomoku side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU050-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in I13\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU051",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU051-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in d1 or h5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU052",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU052-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in f5 or f1\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU053",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU053-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in d4 or h4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU054",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU054-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in e3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU055",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU055-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in g3 or c7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU056",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU056-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in h2 or d6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU057",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU057-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in h6 or d2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU058",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU058-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in b2 or f2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU059",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU059-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in g2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU060",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU060-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in e4 or e8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU062",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU062-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in e9\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU063",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU063-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in c4 or g8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU064",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU064-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in h4 or d8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU065",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU065-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in b3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU066",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU066-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in i1\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU067",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU067-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in e3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU068",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU068-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in d3 or h7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU069",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU069-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in h6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU070",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU070-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in c6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU071",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU071-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in d6 or h2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU072",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU072-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in i3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU073",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU073-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in d8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU074",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The multi-colored checkers side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the multi-colored checkers piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU074-1-1.png"
    ],
    "ground_truth": {
      "answer": "The multi-colored checkers piece should be placed in d5 or h5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU075",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': One side plays using multi-colored checkers pieces, while the other side plays using black and red Chinese chess pieces. The black and red Chinese chess side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black and red Chinese chess piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU075-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black and red Chinese chess piece should be placed in d5 or h5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU076",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using green and orange checkers pieces. The green side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the orange piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU076-1-1.png"
    ],
    "ground_truth": {
      "answer": "The orange piece should be placed in f2 or b6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU077",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using green and orange checkers pieces. The green side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU077-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in b4 or f8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU078",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using green and orange checkers pieces. The green side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU078-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in d3 or d7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU079",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using green and orange checkers pieces. The green side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the orange piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU079-1-1.png"
    ],
    "ground_truth": {
      "answer": "The orange piece should be placed in c2 or g6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU080",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using red and purple checkers pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the purple piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU080-1-1.png"
    ],
    "ground_truth": {
      "answer": "The purple piece should be placed in c6 or g6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU081",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using red and purple checkers pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the purple piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU081-1-1.png"
    ],
    "ground_truth": {
      "answer": "The purple piece should be placed in d3 or d7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU082",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using red and purple checkers pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the purple piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU082-1-1.png"
    ],
    "ground_truth": {
      "answer": "The purple piece should be placed in d2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU083",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using red and purple checkers pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU083-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in h7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU084",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using red and purple checkers pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU084-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in c7 or g7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU085",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using blue and yellow checkers pieces. The blue side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the blue piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU085-1-1.png"
    ],
    "ground_truth": {
      "answer": "The blue piece should be placed in c2 or g6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU086",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using blue and yellow checkers pieces. The blue side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the yellow piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU086-1-1.png"
    ],
    "ground_truth": {
      "answer": "The yellow piece should be placed in d8 or h4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU087",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using blue and yellow checkers pieces. The blue side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the blue piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU087-1-1.png"
    ],
    "ground_truth": {
      "answer": "The blue piece should be placed in g7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU088",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using blue and green checkers pieces. The blue side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU088-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in c5 or g5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU089",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using blue and green checkers pieces. The blue side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU089-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in c3 or g7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU090",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using blue and green checkers pieces. The blue side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the blue piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU090-1-1.png"
    ],
    "ground_truth": {
      "answer": "The blue piece should be placed in d2 or h6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU091",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and orange checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the orange piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU091-1-1.png"
    ],
    "ground_truth": {
      "answer": "The orange piece should be placed in c2 or g6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU092",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and orange checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the orange piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU092-1-1.png"
    ],
    "ground_truth": {
      "answer": "The orange piece should be placed in h3 or d7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU093",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using yellow and green checkers pieces. The yellow side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU093-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in g3 or c7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU094",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using yellow and green checkers pieces. The yellow side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the yellow piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU094-1-1.png"
    ],
    "ground_truth": {
      "answer": "The yellow piece should be placed in g4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU095",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and green checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU095-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in d4 or h8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU096",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and green checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the purple piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU096-1-1.png"
    ],
    "ground_truth": {
      "answer": "The purple piece should be placed in c4 or g4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU097",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and green checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU097-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in g5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU098",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and green checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU098-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in i9\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU099",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and green checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU099-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in d6 or h6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU100",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using purple and green checkers pieces. The purple side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the green piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU100-1-1.png"
    ],
    "ground_truth": {
      "answer": "The green piece should be placed in c6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU101",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU101-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in e4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU102",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU102-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in i8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU103",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU103-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in i4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU104",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU104-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in c5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU105",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU105-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in a4\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU107",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU107-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in b9\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU108",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU108-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in a8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU110",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU110-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in c1\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU111",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The black side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU111-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in a1\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU112",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU112-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in f6\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU113",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU113-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in c4 or g8\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU114",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU114-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in h9\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU115",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU115-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in i0\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU116",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU116-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in h0\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU117",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU117-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in e5\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU118",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU118-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in i3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU119",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU119-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in b3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU120",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU120-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in h7\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU121",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the black piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU121-1-1.png"
    ],
    "ground_truth": {
      "answer": "The black piece should be placed in f3\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU122",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU122-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in g1\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU123",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU123-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in d0\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU124",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU124-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in c2\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  },
  {
    "id": "Geometric_Reasoning/Gomoku_Variation/GOMOKU125",
    "system_prompt": "",
    "prompt": "Here are the rules of 'New Gomoku': Both sides play using black and red Chinese chess pieces. The red side plays first, and then players take turns placing their pieces on the grid of the board. The player who forms five connected pieces first wins. Please determine where the red piece should be placed in the game shown in the image. Use coordinates in the format of a letter (column) + a number (row), such as A8.",
    "image_list": [
      "Gomoku_Variation/GOMOKU125-1-1.png"
    ],
    "ground_truth": {
      "answer": "The red piece should be placed in c0\n\n## Scoring Criteria\nIf the reference answer specifies a single move position, score 1 point when the student's answer matches the specified position exactly (case-insensitive); otherwise, score 0 points.\nIf the reference answer specifies multiple move positions:\n- Score 1 point if all move positions provided in the student's answer are included in the reference answer.\n- Score 0 points if the student's answer includes any move position that is not in the reference answer.\n- Score 0 points if the student's answer does not include any of the move positions listed in the reference answer.",
      "rationale": ""
    },
    "judge_system_prompt": "You are a grading teacher tasked with reviewing and scoring student answers based on the reference answer. During the grading process, you must adhere to the following important points:\n\n- The scoring is based solely on the correctness of the student’s final answer compared to the reference answer. There is no need to assess whether the intermediate steps in the solution are correct.  \n- First, extract the final answer provided by the student and display it in your analysis result. Then, judge the correctness of the extracted answer based on the reference answer.  \n- Assign a score based on your analysis. When explaining the scoring analysis, the explanation should be broken down logically into sections. At the end of your explanation, summarize the analysis and format it as: \"In conclusion, the student’s answer should receive x points\" (where x indicates the specific score awarded).  \n- Keep your explanation concise, limited to 200 words.  \n- Provide the final score in \"JSON\" format using a code block.\n\nYour output format should be:  \n[Scoring analysis]:  \n[Score]: x points  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[score]]\n}\n```\n\n### Scoring Criteria:\n[Score Levels]:\nThe final answer is assessed according to the reference answer key and assigned one of two levels:  \n- 1 Point: Maximum score.  \n- 0 Points: Minimum score. (Even if the answer would merit less than 0 points, apply 0 as the lowest score.)  \n\n[Detailed Explanation]:\n- 1 Point:  \n  - The student's final answer matches the reference answer exactly.  \n  - For questions with multiple subparts, all subparts must be correct to receive 1 point.  \n  - If the student’s answer is mathematically equivalent to the reference answer (e.g., if the student writes 1+1/2x while the reference answer is 1+0.5x, this is acceptable as 1/2=0.5 and earns full credit.  \n- 0 Points:\n  - The student’s final answer does not match the reference answer.  \n  - The student’s answer is empty.  \n\nExample 1:  \n\n<Question>:  \nXiao Ming decides to host a party. If he has 5 tables and each table is paired with 10 chairs, how many more chairs than tables does he have?  \n\n<Reference Answer>: 45  \n\n<Student’s Answer>:  \nXiao Ming has $5$ tables, and each table is paired with $10$ chairs. So, he has a total of $5\\times10=50$ chairs.  \nThe number of chairs minus the number of tables is $50-5=45$.  \nThus, Xiao Ming has $45$ more chairs than tables.  \n\n[Scoring analysis]:  \nThe student’s final answer is that there are 45 more chairs than tables.  \nThe reference answer is 45.  \nThe two answers are consistent, so the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 2:  \n\n<Question>:  \nBox A contains 42 oranges, and Box B contains 18 oranges. If the same number of oranges is removed from both boxes such that the remaining oranges in Box A are 9 times those in Box B, how many oranges remain in each box?  \n\n<Reference Answer>: 27, 3  \n\n<Student’s Answer>:  \nLet the number of oranges removed from each box be \\(x\\).  \nAccording to the problem, the remaining oranges in Box A are 9 times those in Box B. This gives the equation:  \n\n(42 - x) = (18 - x) * 9  \n\nExpanding the equation:  \n42 - x = 162 - 9x.  \n\nRearranging terms:  \n9x - x = 162 - 42.\n\nSimplifying:  \n8x = 120.  \n\nSolving for x:  \nx = 120 / 8  \nx = 15.  \n\nNow, calculate the remaining oranges:  \nBox A: 42 - 15 = 27,  \nBox B: 18 - 15 = 3.  \n\nVerify the result:  \nCheck whether the remaining oranges in Box A are 9 times those in Box B:  \n27 = 3 * 9\n27 = 27\n\nThus, the solution satisfies the problem conditions. The final answer is:  \nBox A has 27 oranges remaining, and Box B has 3 oranges remaining.  \n\n[Scoring analysis]:  \nThe student’s final answers are 27 for Box A and 3 for Box B.  \nThe reference answers are 27 and 3.  \nThe student’s answers match the reference answers. Therefore, the student earns 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```\n\nExample 3:  \n\n<Question>:  \nWhat is the value of  sin(105°)?  \n\n<Reference Answer>: \\(\\frac{\\sqrt{2} + \\sqrt{6}}{4}\\)  \n\n<Student’s Answer>:  \nTo calculate sin(105°), we can use the trigonometric addition formula and properties of angles greater than 90°. 105° can be expressed as 45° + 60°.  \nThe formula for sine of a sum is:  \nsin(A + B) = sin(A)cos(B) + cos(A)sin(B).  \n\nThus:  \nsin(105°) = sin(45° + 60°)\n= sin(45°)cos(60°) + cos(45°)sin(60°)\nWe know:  \nsin(45°) = cos(45°) = √2/2\nsin(60°) = √3/2\ncos(60°) = 1/2\nSubstituting these values:  \nsin(105°) = (√2/2)(1/2) + (√2/2)(√3/2)\n= √2/4 + √6/4\n= (√2 + √6) / 4.  \n\nThus,  sin(105°) 's exact value is  (√2 + √6) / 4.  \n\n[Scoring analysis]:  \nThe student’s final answer is (√2 + √6) / 4.  \nThe reference answer is \\\\frac{\\\\sqrt6+\\\\sqrt2}{4}, expressed in LaTeX format. Both answers are mathematically equivalent as\\\\frac{\\\\sqrt6+\\\\sqrt2}{4}=(√6 + √2)/4=(√2 + √6)/4.  \nTherefore, the student’s answer is correct, and they earn 1 point.  \n\n[Score]: 1 point  \n[JSON]:  \n```json\n{\n  \"answer_score\": [[1]]\n}\n```",
    "judge_prompt_template": "<Question>:\n{prompt}\n\n<Reference Answer>:\n{response_reference}\n\n<Student’s Answer>:\n{response}",
    "extra": {
      "difficulty": "",
      "Category": "Geometric Reasoning",
      "Subtask": "Gomoku Variation",
      "tags": [],
      "metadata": {
        "source": ""
      }
    }
  }
]