problem
string | answer
string | init_response
string | init_reward
int64 | success_rate
float64 | progress
string | action
string | prefix
string | tokenized_conv
string | responses
sequence | rewrads
sequence | mean_reward
float64 |
|---|---|---|---|---|---|---|---|---|---|---|---|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
3 / 66
|
Continue
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["But since the upper limit for each a_i is the previous one, the sequence is non-increasing. So for(...TRUNCATED)
|
[
0,
1,
1,
1,
1,
1,
0,
1,
1,
1,
1,
0,
1,
1,
1,
1
] | 0.8125
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
3 / 66
|
Terminate
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["\n\nTo evaluate the sum:\n\n$$\nN = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} (...TRUNCATED)
|
[
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0.0625
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
8 / 66
|
Continue
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["But how can I compute this?\n\nAlternatively, maybe there's a generating function approach here. L(...TRUNCATED)
|
[
1,
1,
0,
1,
1,
1,
0,
1,
1,
0,
0,
1,
1,
1,
1,
1
] | 0.75
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
8 / 66
|
Terminate
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["\n\nTo evaluate the given nested sum:\n\n$$\nN = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_(...TRUNCATED)
|
[
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1
] | 0.0625
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
15 / 66
|
Continue
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["But I need to think about how to model the sum.\n\nAlternatively, maybe I can find a recurrence re(...TRUNCATED)
|
[
1,
1,
1,
0,
1,
1,
1,
1,
1,
1,
0,
1,
1,
1,
1,
1
] | 0.875
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
15 / 66
|
Terminate
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["\n\nTo evaluate the sum\n\n$$\nN = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \(...TRUNCATED)
|
[
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
28 / 66
|
Continue
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["Alternatively, maybe the pattern is S(k) = 2^{k+1} - 1. Let me check for k=1, 2, 3.\n\nFor k=1: 2^(...TRUNCATED)
|
[
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
0,
1,
1
] | 0.9375
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
28 / 66
|
Terminate
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["\n\nTo evaluate the given nested sum:\n\n$$\nN = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_(...TRUNCATED)
|
[
0,
0,
0,
0,
1,
1,
0,
0,
1,
1,
0,
1,
1,
0,
0,
0
] | 0.375
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
35 / 66
|
Continue
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["But if we consider that for each sequence of length k, the product is a_1 * a_2 * ... * a_k. If we(...TRUNCATED)
|
[
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
] | 1
|
"Let\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} =(...TRUNCATED)
|
95
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| 1
| 0.95
|
35 / 66
|
Terminate
| "<think>\nOkay, so I need to find the value of this big nested sum N, which is a sum over a bunch of(...TRUNCATED)
| "<|im_start|>user\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots(...TRUNCATED)
| ["\n\nTo evaluate the sum:\n\n$$\nN = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} (...TRUNCATED)
|
[
1,
0,
0,
1,
1,
0,
0,
0,
0,
1,
1,
0,
0,
0,
1,
0
] | 0.375
|
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