id stringlengths 20 32 | task stringclasses 53
values | source stringclasses 53
values | split stringclasses 1
value | problem stringlengths 227 504 | prompt stringlengths 246 523 | answer stringlengths 1 40 | ref stringlengths 1 40 | family stringclasses 53
values | domain stringclasses 7
values | difficulty int64 0 3 | tier int64 0 3 | rung stringclasses 1
value | n_valid int64 2 64 | bucket stringclasses 3
values | guess_rate float64 0 0.02 | solvable bool 1
class | holdout bool 1
class | meta dict | cell stringclasses 212
values | stage stringclasses 3
values |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
po-ap_sum-t0-rl-000000 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -9, 9 | -9, 9 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 81
},
"seed": 141063620200000000,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 5,
"valid_answers": [
"-9, 9",
"-4, 7",
"1, 5",
"6, 3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000001 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 111. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 111. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -4, 9 | -4, 9 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 111
},
"seed": 141063620200000000,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 6,
"valid_answers": [
"-4, 9",
"1, 7",
"6, 5"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000002 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, 2 | -9, 2 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -24
},
"seed": 141063620200000000,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 2,
"valid_answers": [
"-9, 2",
"-4, 0",
"1, -2",
"6, -4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000004 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, 2 | -8, 2 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -18
},
"seed": 141063620200000000,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 2,
"valid_answers": [
"-8, 2",
"-3, 0",
"2, -2",
"7, -4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000005 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -76. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -76. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, -6 | -10, -6 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -76
},
"seed": 141063620200000000,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 8,
"valid_answers": [
"-10, -6",
"-7, -8",
"-4, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000006 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -10, 10 | -10, 10 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 90
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-10, 10",
"-5, 8",
"0, 6",
"5, 4",
"10, 2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000007 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -3, 10 | -3, 10 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 48
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 6,
"valid_answers": [
"-3, 10",
"0, 8",
"3, 6",
"6, 4",
"9, 2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000008 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 51. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 51. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -9, 7 | -9, 7 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 51
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 3,
"valid_answers": [
"-9, 7",
"-4, 5",
"1, 3",
"6, 1"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000009 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -10, 6 | -10, 6 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 30
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 2,
"valid_answers": [
"-10, 6",
"-5, 4",
"0, 2",
"5, 0",
"10, -2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000010 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -10, -5 | -10, -5 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": -100
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 7,
"valid_answers": [
"-10, -5",
"-8, -6",
"-6, -7",
"-4, -8",
"-2, -9",
"0, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000011 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 132. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -3, 10 | -3, 10 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 132
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 7,
"valid_answers": [
"-3, 10",
"2, 8",
"7, 6"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000012 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -38. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, -1 | -8, -1 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -38
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-8, -1",
"-5, -3",
"-2, -5",
"1, -7",
"4, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000013 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -8, 5 | -8, 5 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -2
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 1,
"valid_answers": [
"-8, 5",
"-5, 3",
"-2, 1",
"1, -1",
"4, -3",
"7, -5",
"10, -7"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000014 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the value... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by ... | -9, 4 | -9, 4 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 6
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 1,
"valid_answers": [
"-9, 4",
"-4, 2",
"1, 0",
"6, -2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000015 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -9. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -9, 3 | -9, 3 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -9
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 1,
"valid_answers": [
"-9, 3",
"-4, 1",
"1, -1",
"6, -3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000016 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 46. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -2, 9 | -2, 9 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 46
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-2, 9",
"1, 7",
"4, 5",
"7, 3",
"10, 1"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000017 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -9, 8 | -9, 8 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 12
},
"seed": 141063620200000020,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-9, 8",
"-6, 6",
"-3, 4",
"0, 2",
"3, 0",
"6, -2",
"9, -4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000018 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -45. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 1 | -10, 1 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -45
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 3,
"valid_answers": [
"-10, 1",
"-5, -1",
"0, -3",
"5, -5",
"10, -7"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000019 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -9, 5 | -9, 5 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -6
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 1,
"valid_answers": [
"-9, 5",
"-6, 3",
"-3, 1",
"0, -1",
"3, -3",
"6, -5",
"9, -7"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000020 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -2, 10 | -2, 10 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": 90
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 6,
"valid_answers": [
"-2, 10",
"0, 9",
"2, 8",
"4, 7",
"6, 6",
"8, 5",
"10, 4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000021 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -105. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -105. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -10, -3 | -10, -3 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -105
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 5,
"valid_answers": [
"-10, -3",
"-5, -5",
"0, -7",
"5, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000022 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 138. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 138. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -2, 10 | -2, 10 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 138
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 8,
"valid_answers": [
"-2, 10",
"3, 8",
"8, 6"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000023 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -6, -3 | -6, -3 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -81
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 5,
"valid_answers": [
"-6, -3",
"-1, -5",
"4, -7",
"9, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000024 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 3 | -10, 3 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -22
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 3,
"valid_answers": [
"-10, 3",
"-7, 1",
"-4, -1",
"-1, -3",
"2, -5",
"5, -7",
"8, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000025 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the value... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by ... | -10, 8 | -10, 8 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 8
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-10, 8",
"-7, 6",
"-4, 4",
"-1, 2",
"2, 0",
"5, -2",
"8, -4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000026 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -129. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -129. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -9, -5 | -9, -5 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -129
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 7,
"valid_answers": [
"-9, -5",
"-4, -7",
"1, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000028 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -10, 8 | -10, 8 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 60
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 4,
"valid_answers": [
"-10, 8",
"-5, 6",
"0, 4",
"5, 2",
"10, 0"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000029 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -6, 6 | -6, 6 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 54
},
"seed": 141063620200000030,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 4,
"valid_answers": [
"-6, 6",
"-1, 4",
"4, 2",
"9, 0"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000030 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 108. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 108. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -7, 10 | -7, 10 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 108
},
"seed": 141063620200000050,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-7, 10",
"-2, 8",
"3, 6",
"8, 4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000031 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -95. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -95. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -5 | -9, -5 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": -95
},
"seed": 141063620200000050,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 7,
"valid_answers": [
"-9, -5",
"-7, -6",
"-5, -7",
"-3, -8",
"-1, -9",
"1, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000032 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -9, -6 | -9, -6 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -144
},
"seed": 141063620200000050,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 8,
"valid_answers": [
"-9, -6",
"-4, -8",
"1, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000033 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -7 | -9, -7 | po_ap_sum | sequence | 0 | 0 | witness | 2 | 2-3 | 0.004535 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -78
},
"seed": 141063620200000050,
"version": "po-v0",
"tier": 0,
"n_raw": 2,
"small": 9,
"valid_answers": [
"-9, -7",
"-6, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000035 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 15. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -10, 5 | -10, 5 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 15
},
"seed": 141063620200000050,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 1,
"valid_answers": [
"-10, 5",
"-5, 3",
"0, 1",
"5, -1",
"10, -3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000036 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -62. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -62. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, -5 | -8, -5 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -62
},
"seed": 141063620200000050,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 7,
"valid_answers": [
"-8, -5",
"-5, -7",
"-2, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000037 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 48. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -7, 6 | -7, 6 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 48
},
"seed": 141063620200000050,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 3,
"valid_answers": [
"-7, 6",
"-2, 4",
"3, 2",
"8, 0"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000040 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | 6, 9 | 6, 9 | po_ap_sum | sequence | 0 | 0 | witness | 2 | 2-3 | 0.004535 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 78
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 2,
"small": 9,
"valid_answers": [
"6, 9",
"9, 7"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000041 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -50. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -50. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, -3 | -8, -3 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -50
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 5,
"valid_answers": [
"-8, -3",
"-5, -5",
"-2, -7",
"1, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000042 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the value... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by ... | -9, 7 | -9, 7 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 6
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 1,
"valid_answers": [
"-9, 7",
"-6, 5",
"-3, 3",
"0, 1",
"3, -1",
"6, -3",
"9, -5"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000043 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 140. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 140. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | 8, 10 | 8, 10 | po_ap_sum | sequence | 0 | 0 | witness | 2 | 2-3 | 0.004535 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": 140
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 2,
"small": 10,
"valid_answers": [
"8, 10",
"10, 9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000044 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -40. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 0 | -10, 0 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -40
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 4,
"valid_answers": [
"-10, 0",
"-7, -2",
"-4, -4",
"-1, -6",
"2, -8",
"5, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000045 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -6. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -6, 2 | -6, 2 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -6
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 1,
"valid_answers": [
"-6, 2",
"-1, 0",
"4, -2",
"9, -4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000047 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -30. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 2 | -10, 2 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -30
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 2,
"valid_answers": [
"-10, 2",
"-5, 0",
"0, -2",
"5, -4",
"10, -6"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000049 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -52. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -52. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, -2 | -10, -2 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -52
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 6,
"valid_answers": [
"-10, -2",
"-7, -4",
"-4, -6",
"-1, -8",
"2, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000050 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -7, 4 | -7, 4 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 18
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 2,
"valid_answers": [
"-7, 4",
"-2, 2",
"3, 0",
"8, -2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000051 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 52. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 52. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -2, 10 | -2, 10 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 52
},
"seed": 141063620200000060,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 6,
"valid_answers": [
"-2, 10",
"1, 8",
"4, 6",
"7, 4",
"10, 2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000052 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 44. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -4, 10 | -4, 10 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 44
},
"seed": 141063620200000080,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-4, 10",
"-1, 8",
"2, 6",
"5, 4",
"8, 2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000053 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -84. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -84. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -8 | -9, -8 | po_ap_sum | sequence | 0 | 0 | witness | 2 | 2-3 | 0.004535 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -84
},
"seed": 141063620200000080,
"version": "po-v0",
"tier": 0,
"n_raw": 2,
"small": 9,
"valid_answers": [
"-9, -8",
"-6, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000054 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 102. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 102. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, 10 | -8, 10 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 102
},
"seed": 141063620200000080,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-8, 10",
"-3, 8",
"2, 6",
"7, 4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000055 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -9, 6 | -9, 6 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 36
},
"seed": 141063620200000080,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 2,
"valid_answers": [
"-9, 6",
"-4, 4",
"1, 2",
"6, 0"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000057 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -8, 8 | -8, 8 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 16
},
"seed": 141063620200000080,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-8, 8",
"-5, 6",
"-2, 4",
"1, 2",
"4, 0",
"7, -2",
"10, -4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000058 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 2 | -10, 2 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -28
},
"seed": 141063620200000080,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 4,
"valid_answers": [
"-10, 2",
"-7, 0",
"-4, -2",
"-1, -4",
"2, -6",
"5, -8",
"8, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000059 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -36. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, 0 | -9, 0 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -36
},
"seed": 141063620200000080,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 4,
"valid_answers": [
"-9, 0",
"-6, -2",
"-3, -4",
"0, -6",
"3, -8",
"6, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000061 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -69. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -69. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -1 | -9, -1 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -69
},
"seed": 141063620200000100,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 4,
"valid_answers": [
"-9, -1",
"-4, -3",
"1, -5",
"6, -7"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000062 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 72. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 72. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -8, 8 | -8, 8 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 72
},
"seed": 141063620200000100,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 4,
"valid_answers": [
"-8, 8",
"-3, 6",
"2, 4",
"7, 2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000064 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -90. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, -2 | -10, -2 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -90
},
"seed": 141063620200000100,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-10, -2",
"-5, -4",
"0, -6",
"5, -8",
"10, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000065 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 1 | -10, 1 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -34
},
"seed": 141063620200000100,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 4,
"valid_answers": [
"-10, 1",
"-7, -1",
"-4, -3",
"-1, -5",
"2, -7",
"5, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000066 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -10, 9 | -10, 9 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 75
},
"seed": 141063620200000100,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-10, 9",
"-5, 7",
"0, 5",
"5, 3",
"10, 1"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000067 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -123. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -123. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -8, -5 | -8, -5 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -123
},
"seed": 141063620200000100,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 7,
"valid_answers": [
"-8, -5",
"-3, -7",
"2, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000069 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -8. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -8, 4 | -8, 4 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -8
},
"seed": 141063620200000110,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-8, 4",
"-5, 2",
"-2, 0",
"1, -2",
"4, -4",
"7, -6",
"10, -8"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000070 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 110. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 110. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | 2, 10 | 2, 10 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": 110
},
"seed": 141063620200000110,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 8,
"valid_answers": [
"2, 10",
"4, 9",
"6, 8",
"8, 7",
"10, 6"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000072 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 95. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 95. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -1, 10 | -1, 10 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": 95
},
"seed": 141063620200000110,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 7,
"valid_answers": [
"-1, 10",
"1, 9",
"3, 8",
"5, 7",
"7, 6",
"9, 5"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000073 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to -75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -3 | -9, -3 | po_ap_sum | sequence | 0 | 0 | witness | 8 | 4-8 | 0.018141 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": -75
},
"seed": 141063620200000110,
"version": "po-v0",
"tier": 0,
"n_raw": 8,
"small": 5,
"valid_answers": [
"-9, -3",
"-7, -4",
"-5, -5",
"-3, -6",
"-1, -7",
"1, -8",
"3, -9",
"... | ap_sum/0 | B |
po-ap_sum-t0-rl-000075 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -60. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 0 | -10, 0 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -60
},
"seed": 141063620200000110,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 4,
"valid_answers": [
"-10, 0",
"-5, -2",
"0, -4",
"5, -6",
"10, -8"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000078 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -8, 5 | -8, 5 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 27
},
"seed": 141063620200000130,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 2,
"valid_answers": [
"-8, 5",
"-3, 3",
"2, 1",
"7, -1"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000080 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | 1, 9 | 1, 9 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 58
},
"seed": 141063620200000130,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 7,
"valid_answers": [
"1, 9",
"4, 7",
"7, 5",
"10, 3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000082 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -12. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, 4 | -9, 4 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -12
},
"seed": 141063620200000130,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-9, 4",
"-6, 2",
"-3, 0",
"0, -2",
"3, -4",
"6, -6",
"9, -8"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000083 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -32. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, 0 | -8, 0 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -32
},
"seed": 141063620200000130,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 4,
"valid_answers": [
"-8, 0",
"-5, -2",
"-2, -4",
"1, -6",
"4, -8",
"7, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000085 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -84. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -84. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -2 | -9, -2 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -84
},
"seed": 141063620200000130,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 4,
"valid_answers": [
"-9, -2",
"-4, -4",
"1, -6",
"6, -8"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000086 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -27. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -7, 1 | -7, 1 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -27
},
"seed": 141063620200000140,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 2,
"valid_answers": [
"-7, 1",
"-2, -1",
"3, -3",
"8, -5"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000087 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -147. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -147. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -7, -7 | -7, -7 | po_ap_sum | sequence | 0 | 0 | witness | 2 | 2-3 | 0.004535 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -147
},
"seed": 141063620200000140,
"version": "po-v0",
"tier": 0,
"n_raw": 2,
"small": 7,
"valid_answers": [
"-7, -7",
"-2, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000088 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -120. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -10, -4 | -10, -4 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -120
},
"seed": 141063620200000140,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-10, -4",
"-5, -6",
"0, -8",
"5, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000092 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 105. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 105. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -5, 9 | -5, 9 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 105
},
"seed": 141063620200000160,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 5,
"valid_answers": [
"-5, 9",
"0, 7",
"5, 5",
"10, 3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000093 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the value... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 0. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by ... | -9, 6 | -9, 6 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 0
},
"seed": 141063620200000160,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 0,
"valid_answers": [
"-9, 6",
"-6, 4",
"-3, 2",
"0, 0",
"3, -2",
"6, -4",
"9, -6"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000094 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 56. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 56. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -1, 10 | -1, 10 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 56
},
"seed": 141063620200000160,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-1, 10",
"2, 8",
"5, 6",
"8, 4"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000095 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the value... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 2. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by ... | -10, 7 | -10, 7 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 2
},
"seed": 141063620200000160,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 1,
"valid_answers": [
"-10, 7",
"-7, 5",
"-4, 3",
"-1, 1",
"2, -1",
"5, -3",
"8, -5"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000096 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 28. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -8, 10 | -8, 10 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 28
},
"seed": 141063620200000180,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 4,
"valid_answers": [
"-8, 10",
"-5, 8",
"-2, 6",
"1, 4",
"4, 2",
"7, 0",
"10, -2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000098 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -16. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, 4 | -10, 4 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -16
},
"seed": 141063620200000180,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-10, 4",
"-7, 2",
"-4, 0",
"-1, -2",
"2, -4",
"5, -6",
"8, -8"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000099 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -56. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -56. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, -4 | -8, -4 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -56
},
"seed": 141063620200000180,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-8, -4",
"-5, -6",
"-2, -8",
"1, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000101 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -54. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -3 | -9, -3 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -54
},
"seed": 141063620200000200,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-9, -3",
"-6, -5",
"-3, -7",
"0, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000102 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -18. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, 3 | -9, 3 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -18
},
"seed": 141063620200000200,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 3,
"valid_answers": [
"-9, 3",
"-6, 1",
"-3, -1",
"0, -3",
"3, -5",
"6, -7",
"9, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000103 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 14. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -10, 9 | -10, 9 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 14
},
"seed": 141063620200000200,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-10, 9",
"-7, 7",
"-4, 5",
"-1, 3",
"2, 1",
"5, -1",
"8, -3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000104 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -64. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -64. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, -4 | -10, -4 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -64
},
"seed": 141063620200000200,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 7,
"valid_answers": [
"-10, -4",
"-7, -6",
"-4, -8",
"-1, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000106 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 135. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 135. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | 7, 10 | 7, 10 | po_ap_sum | sequence | 0 | 0 | witness | 2 | 2-3 | 0.004535 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": 135
},
"seed": 141063620200000200,
"version": "po-v0",
"tier": 0,
"n_raw": 2,
"small": 9,
"valid_answers": [
"7, 10",
"9, 9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000107 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -75. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, -1 | -10, -1 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -75
},
"seed": 141063620200000200,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-10, -1",
"-5, -3",
"0, -5",
"5, -7",
"10, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000110 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 105. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 105. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | 1, 10 | 1, 10 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": 105
},
"seed": 141063620200000210,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 7,
"valid_answers": [
"1, 10",
"3, 9",
"5, 8",
"7, 7",
"9, 6"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000111 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -20. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -8, 2 | -8, 2 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -20
},
"seed": 141063620200000210,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 2,
"valid_answers": [
"-8, 2",
"-5, 0",
"-2, -2",
"1, -4",
"4, -6",
"7, -8",
"10, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000113 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -96. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -6, -4 | -6, -4 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -96
},
"seed": 141063620200000210,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-6, -4",
"-1, -6",
"4, -8",
"9, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000114 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 24. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -9, 10 | -9, 10 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 24
},
"seed": 141063620200000220,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 3,
"valid_answers": [
"-9, 10",
"-6, 8",
"-3, 6",
"0, 4",
"3, 2",
"6, 0",
"9, -2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000115 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 5 terms equal to 100. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | 0, 10 | 0, 10 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 5,
"B": 10,
"S": 100
},
"seed": 141063620200000220,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 7,
"valid_answers": [
"0, 10",
"2, 9",
"4, 8",
"6, 7",
"8, 6",
"10, 5"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000116 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 70. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 70. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | 4, 9 | 4, 9 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 70
},
"seed": 141063620200000220,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 7,
"valid_answers": [
"4, 9",
"7, 7",
"10, 5"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000117 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -114. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the va... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to -114. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced ... | -9, -4 | -9, -4 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": -114
},
"seed": 141063620200000220,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 6,
"valid_answers": [
"-9, -4",
"-4, -6",
"1, -8",
"6, -10"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000118 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 165. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 165. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | 5, 9 | 5, 9 | po_ap_sum | sequence | 0 | 0 | witness | 2 | 2-3 | 0.004535 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 165
},
"seed": 141063620200000220,
"version": "po-v0",
"tier": 0,
"n_raw": 2,
"small": 9,
"valid_answers": [
"5, 9",
"10, 7"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000119 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 144. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -1, 10 | -1, 10 | po_ap_sum | sequence | 0 | 0 | witness | 3 | 2-3 | 0.006803 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 144
},
"seed": 141063620200000240,
"version": "po-v0",
"tier": 0,
"n_raw": 3,
"small": 8,
"valid_answers": [
"-1, 10",
"4, 8",
"9, 6"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000120 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 78. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -7, 8 | -7, 8 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 78
},
"seed": 141063620200000240,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 4,
"valid_answers": [
"-7, 8",
"-2, 6",
"3, 4",
"8, 2"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000121 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 39. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -6, 5 | -6, 5 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 39
},
"seed": 141063620200000240,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 3,
"valid_answers": [
"-6, 5",
"-1, 3",
"4, 1",
"9, -1"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000122 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 22. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -8, 9 | -8, 9 | po_ap_sum | sequence | 0 | 0 | witness | 7 | 4-8 | 0.015873 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 22
},
"seed": 141063620200000240,
"version": "po-v0",
"tier": 0,
"n_raw": 7,
"small": 3,
"valid_answers": [
"-8, 9",
"-5, 7",
"-2, 5",
"1, 3",
"4, 1",
"7, -1",
"10, -3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000123 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 93. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -7, 9 | -7, 9 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 6,
"B": 10,
"S": 93
},
"seed": 141063620200000240,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 5,
"valid_answers": [
"-7, 9",
"-2, 7",
"3, 5",
"8, 3"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000124 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -58. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -10, -3 | -10, -3 | po_ap_sum | sequence | 0 | 0 | witness | 4 | 4-8 | 0.00907 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -58
},
"seed": 141063620200000260,
"version": "po-v0",
"tier": 0,
"n_raw": 4,
"small": 7,
"valid_answers": [
"-10, -3",
"-7, -5",
"-4, -7",
"-1, -9"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000125 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the valu... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to 34. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by... | -5, 9 | -5, 9 | po_ap_sum | sequence | 0 | 0 | witness | 6 | 4-8 | 0.013605 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": 34
},
"seed": 141063620200000260,
"version": "po-v0",
"tier": 0,
"n_raw": 6,
"small": 4,
"valid_answers": [
"-5, 9",
"-2, 7",
"1, 5",
"4, 3",
"7, 1",
"10, -1"
]
} | ap_sum/0 | B |
po-ap_sum-t0-rl-000126 | cx_po_ap_sum | cx_po_ap_sum | rl | Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the val... | Problem:
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 4 terms equal to -42. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced b... | -9, -1 | -9, -1 | po_ap_sum | sequence | 0 | 0 | witness | 5 | 4-8 | 0.011338 | true | false | {
"family": "po_ap_sum",
"rung": "witness",
"params": {
"n": 4,
"B": 10,
"S": -42
},
"seed": 141063620200000260,
"version": "po-v0",
"tier": 0,
"n_raw": 5,
"small": 5,
"valid_answers": [
"-9, -1",
"-6, -3",
"-3, -5",
"0, -7",
"3, -9"
]
} | ap_sum/0 | B |
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ultraviolet param-object pools
Synthetic, exactly verifiable math construction problems for RL. Each item asks for a short integer tuple that parameterises a mathematical object (polynomial coefficients, matrix entries, a lattice point, a permutation, ...) with stated properties. Every item has a finite answer box, so all of its valid answers are enumerated exactly and stored in the row; any one valid answer is correct. Each item has 2-64 valid answers and a blind-guess success rate of at most 0.02.
| split | file | rows | md5 |
|---|---|---|---|
| train | train.jsonl |
239,584 | d59f16328d437c68d6b96caf0b0c44a8 |
| eval | eval.jsonl |
2,241 | 6a8c2cf0f50a53ac5ba0223ca74da4c8 |
| eval_tier0 (tier 0 only) | eval_tier0.jsonl |
324 | d4fb551a4c6afd10198c78f0f1460f76 |
The eval splits share no problems with train.
Example
Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its first 6 terms equal to 81. Several answers are possible; any one correct answer is accepted. Give one solution. End with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the values.
Valid answers (meta.valid_answers): -9, 9, -4, 7, 1, 5, 6, 3.
Fields
problem (statement), prompt (Problem:\n{problem}\n\n<think>\n), answer (one valid answer), family, domain,
tier (0-3, difficulty), n_valid, guess_rate, meta.params / meta.seed (the generator's inputs) and
meta.valid_answers.
Usage
from datasets import load_dataset
ds = load_dataset("harribo11/ultraviolet-paramobj-pools", split="train")
Generator, verifier and scorer: workbench/envs/paramobj/ in
github.com/hliuson/ultraviolet. There the tasks are registered as
wb_po_<name>, and the rows here say cx_po_<name>, so rename the task field before scoring:
from ultraviolet.tasks.score import score
row = dict(row, task="wb_" + row["task"][3:]) # cx_po_ap_sum -> wb_po_ap_sum
score(row["task"], completion, row) # 1.0 if the last \boxed{} after </think> is a valid answer
Families (53) and tiers
Train rows per tier, and eval rows per family:
| family | domain | t0 | t1 | t2 | t3 | eval |
|---|---|---|---|---|---|---|
| po_ap_sum | sequence | 186 | 842 | 1,979 | 1,998 | 48 |
| po_binary_runs | combinatorics | 15 | 43 | 73 | 64 | 32 |
| po_circle_through | geometry | 498 | 1,853 | 2,015 | 2,002 | 48 |
| po_consecutive_sum | number | 18 | 117 | 211 | 1,471 | 35 |
| po_crt_range | number | 1,987 | 2,037 | 1,989 | 2,033 | 48 |
| po_diff_squares | number | 26 | 30 | 52 | 119 | 14 |
| po_egyptian | number | 23 | 97 | 74 | 141 | 29 |
| po_fraction_between | number | 1,564 | 1,971 | 1,985 | 2,014 | 48 |
| po_gauss_factor | number | 33 | 166 | 453 | 1,926 | 36 |
| po_gcd_lcm | number | 280 | 774 | 1,170 | 91 | 48 |
| po_gp_term | sequence | 110 | 113 | 433 | 191 | 48 |
| po_int_triangle | geometry | 118 | 503 | 1,981 | 1,988 | 48 |
| po_lattice_tri_area | geometry | 1,611 | 1,956 | 2,001 | 2,009 | 48 |
| po_lin2_box | number | 1,995 | 2,011 | 2,029 | 2,008 | 48 |
| po_lin_commute | function | 756 | 1,599 | 2,008 | 2,018 | 48 |
| po_lin_congruence | number | 1,997 | 1,995 | 2,012 | 1,971 | 48 |
| po_magic3 | combinatorics | 1,990 | 2,007 | 2,002 | 2,013 | 48 |
| po_mat2_commute | linalg | 2,017 | 2,032 | 1,999 | 2,014 | 48 |
| po_mat2_eigvec | linalg | 679 | 963 | 1,149 | 1,293 | 48 |
| po_mat2_idempotent | linalg | 47 | 715 | 1,364 | 2,023 | 37 |
| po_mat2_involution | linalg | 42 | 609 | 1,318 | 1,999 | 44 |
| po_mat2_nilpotent | linalg | 48 | 456 | 846 | 2,011 | 36 |
| po_mat2_trdet | linalg | 181 | 735 | 1,887 | 2,032 | 48 |
| po_mat3_det | linalg | 2,027 | 1,990 | 1,993 | 2,043 | 48 |
| po_mobius | function | 1,014 | 1,466 | 2,000 | 2,007 | 48 |
| po_moment_weights | combinatorics | 107 | 328 | 675 | 928 | 48 |
| po_order_mod | number | 22 | 154 | 528 | 1,680 | 36 |
| po_parallelogram | geometry | 1,987 | 2,021 | 1,987 | 2,007 | 48 |
| po_partition_sq | combinatorics | 366 | 1,623 | 2,004 | 1,981 | 48 |
| po_perm_cycles | combinatorics | 19 | 45 | 45 | 517 | 35 |
| po_perm_descents | combinatorics | 76 | 248 | 248 | 424 | 48 |
| po_perm_inversions | combinatorics | 15 | 40 | 40 | 354 | 45 |
| po_poly_critical | polynomial | 258 | 1,998 | 1,997 | 1,942 | 48 |
| po_poly_integral | polynomial | 1,993 | 2,002 | 2,016 | 1,999 | 48 |
| po_poly_mixed | polynomial | 2,006 | 2,020 | 1,987 | 1,977 | 48 |
| po_poly_roots | polynomial | 32 | 62 | 143 | 68 | 19 |
| po_poly_slope | polynomial | 767 | 1,963 | 2,007 | 2,015 | 48 |
| po_poly_values | polynomial | 1,003 | 2,018 | 2,000 | 1,994 | 48 |
| po_pyth_leg | number | 25 | 80 | 242 | 867 | 29 |
| po_pyth_perim | geometry | 28 | 68 | 189 | 491 | 41 |
| po_quad_rootdiff | polynomial | 52 | 294 | 518 | 882 | 37 |
| po_recur_coef | sequence | 338 | 1,829 | 700 | 394 | 48 |
| po_recur_init | sequence | 1,008 | 1,915 | 2,019 | 1,982 | 48 |
| po_sfft | number | 1,461 | 1,987 | 1,986 | 1,991 | 48 |
| po_sigma_inv | number | 25 | 126 | 357 | 1,248 | 36 |
| po_sqrt_mod | number | 414 | 2,029 | 2,018 | 1,988 | 48 |
| po_stats_multiset | combinatorics | 32 | 103 | 193 | 1,610 | 36 |
| po_tau_range | number | 2,005 | 1,975 | 2,052 | 1,980 | 48 |
| po_totient_inv | number | 14 | 25 | 48 | 88 | 4 |
| po_two_sq_comp | number | 87 | 727 | 1,997 | 2,039 | 48 |
| po_vec_cross | linalg | 2,007 | 1,980 | 2,002 | 1,991 | 48 |
| po_vec_dot | linalg | 989 | 1,980 | 1,992 | 1,997 | 48 |
| po_vieta3 | polynomial | 14 | 84 | 467 | 1,975 | 28 |
| total | 36,412 | 56,804 | 67,480 | 78,888 |
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