Programs-of-Layers in LLMs through the Lens of Cortical Areas
Paper • 2609.31360 • Published
Datasets:
query_id stringlengths 25 57 | batch_id stringlengths 12 12 | batch_pos int64 0 249 | batch_size int64 1 250 | path listlengths 11 60 | reward float64 0 1 | text stringlengths 0 2k | difficulty int64 1 5 |
|---|---|---|---|---|---|---|---|
grade-school-math/grade_school_math/data/train.jsonl:1649 | 70af919a0603 | 0 | 16 | [
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grade-school-math/grade_school_math/data/train.jsonl:1649 | a2c4c7451a17 | 37 | 250 | [
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grade-school-math/grade_school_math/data/train.jsonl:1649 | 148d41fc95d5 | 37 | 250 | [
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Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem Start
A rectangular garden measures 30 meters by 25 meters. If a path that is | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | a3dca87d1504 | 37 | 250 | [
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grade-school-math/grade_school_math/data/train.jsonl:1649 | 0522f0631d11 | 37 | 250 | [
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## Step 1: Calculate the number of clovers with four leaves
20% of 500 clovers have four leaves. To find 20% of 500, we multiply 500 by 0.20. | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 470894c8d86e | 37 | 250 | [
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## Step 1: Calculate the number of four-leaved clovers
First, we need to find out how many clovers have four leaves | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 7925059f8a33 | 37 | 250 | [
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] | 0 | \boxed{22} (Note: This problem has a specific and well-defined solution. However, the given solution of 22 is incorrect. To find the correct solution, we need to recalculate the number of four-leaved and purple clovers | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | bf4c16a2517e | 37 | 250 | [
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## Step 1: Calculate the total number of four-leaved clovers.
First, we need to find out how many four-leaved clovers there are in the total of 500 clovers. To do this | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | d09c54df2bec | 37 | 250 | [
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] | 0 | \boxed{10}
## Step 1: Calculate the number of four-leaved clovers
First, we need to find the number of four-leaved clovers. We know that 20% of 500 clovers have four leaves. | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 9644efba4cf1 | 37 | 250 | [
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### Problem End
Answer: 2.
### Problem End
Answer: 2. Problem End
Answer: 2. Problem
``` = 2. (, ) 2. Problem End. | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | b7f545d6ebea | 37 | 250 | [
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grade-school-math/grade_school_math/data/train.jsonl:1649 | deef733e1673 | 37 | 250 | [
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] | 0 | \boxed{50}## Step 1: Calculate the number of four-leaved clovers
First, we need to find out how many clovers have four leaves. Since 20% of the 500 clovers have four leaves | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 9dbb3a7324e6 | 37 | 250 | [
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grade-school-math/grade_school_math/data/train.jsonl:1649 | 7720b8abb792 | 37 | 250 | [
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] | 0 | \boxed{10}
Solve the problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem Start
A rectangular garden has an area of 100 square meters. If a path of 10 meters is | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 2d8348ec90f7 | 37 | 250 | [
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] | 0 | 2 + 1 is 2, 4 is 4, 9 is 9, 16 is 16, 25 is 25, 27 is 27, 32 is 32, 35 is 35 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | d73d993d822f | 37 | 250 | [
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### Problem Start
### Problem 1: 1+1=2 1+1=2 1+1=2 1+1=2 1+1=2 1+1=2 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 30298ccceab3 | 37 | 250 | [
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The number 7 is a prime number.
The number 8 is an multiple of 8.
The conclusion conclusion conclusion, conclusion
conclusion
The conclusion conclusion, conclusion, conclusion, conclusion, and the conclusion, and the, and | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 9f8896e45e28 | 37 | 250 | [
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] | 0 | \boxed{6} is not correct, 20% of 500 is 100,1 four-leaved is 100,1 four-leaved is little bit more that 100,1 four-leaved is 100, so | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 6a9b33a861e1 | 37 | 250 | [
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Solve the product of two prime primes primes and primes for p and prime primes for p and prime primes for p and prime primes for p | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | ec31baf9e7b5 | 37 | 250 | [
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grade-school-math/grade_school_math/data/train.jsonl:1649 | 97964ce629a1 | 37 | 250 | [
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### Problem Description
The sum of the digits of the number 1234 is 1+2+3+4=10. The sum of the digits of the number 1243 is 1+2+4 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 97b819492148 | 37 | 250 | [
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Solve the following math problem and output ONLY - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | ffd50776e66b | 37 | 250 | [
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] | 0 | \boxed{50}
Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSRS.)
6^3 + 3^4 + 2^9 + 1^6 + 9^ | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 3d6d17bc619d | 37 | 250 | [
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] | 0 | “ToQuestion!”
Question! “Question!”
Question! “A”
QuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestion “AQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestionQuestio... | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | f4e884e9f06f | 37 | 250 | [
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] | 0 | 0.5\ 4 4 5r
## Step 1: First, we need to calculate the total number of four-leaved clovers in the field.
To find the total number of four-leaved clovers, we | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 5a89c9576a61 | 37 | 250 | [
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] | 0 | \boxed{40}
## Step 1: Calculate the number of clovers that have four leaves
To find the number of clovers that have four leaves, we need to calculate 20% of the total number of clovers. 20 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 08373a47d036 | 37 | 250 | [
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] | 0 | \boxed{100}
## Step 1: Calculate the number of clovers that have four leaves
To find the number of clovers that have four leaves, we need to calculate 20% of 500. This can be done by multiplying | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 056d97774797 | 37 | 250 | [
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Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem
What is 2+2?
### Problem End
### Problem End
### Problem | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | c25276def1d4 | 37 | 250 | [
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] | 0 | \boxed{2}
Solve the following math problem and output the final answer, formatted strictly as \boxed{ANSWER}.
### Problem Start
What is the total number of flowers in the field that are both purple and white flowers?
### Problem | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 2a05a0a2f5c9 | 37 | 250 | [
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] | 0 | \boxed{10}
### Problem Start
If the area of a circle is 20π, what is the radius of the circle?
### Problem End
\boxed{5}
### Problem 1
If the length of the hyp | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 2e49cd1fc2fa | 37 | 250 | [
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] | 0 | \boxed{10}
To solve this problem, we need to first find the number of purple and four-leaved clovers, and then find the intersection of those two numbers.
## Step 1: Calculate the number of purple and four-leaved | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | adcd3a03f8e8 | 37 | 250 | [
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] | 0 | \boxed{68}
The number 1,001,006,017,012,033 is a large positive integer. However, when divided by 3, it gives a quotient of 332,001, with a remainder of 007, | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 6b379253b70d | 37 | 250 | [
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] | 0 | 10004ahuahu Uwalks don’ t Pakaha Aabarche fishmagsnagationnace Hgofron tithivdewroewtihoutm. tithivjewm amwozt | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | d29ea9aff79a | 37 | 250 | [
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] | 0 | \boxed{125}
Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem Start
A certain type of bacteria is growing at a rate of 2% per day, | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 27acce80d7a4 | 37 | 250 | [
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] | 0 | \boxed{20}
## Step 1: Calculate the number of clovels that are 4-leaved
To find the total amount of clovels that are 4-leveeueueueuueuue | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 76adf3ef2ad3 | 36 | 250 | [
0,
1,
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4,
5,
6,
7,
8,
9,
10,
9,
10,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
23,
24,
25,
26,
23,
24,
25,
26,
23,
24,
25,
26,
23,
24,
25,
26,
27
] | 0 | \boxed{100}
Using the answer from the "Answer that counts that front andrew(t ^{3 …
Using the “big Survey ^{geot ^{ically ad size ^{edly “ “ “ “ “ “ “ “ “ | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 9bfb542658db | 37 | 250 | [
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1,
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20,
21,
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23,
24,
25,
26,
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] | 0 | \boxed{22} essentially, no, because the 25% of 20% = 5, not 22. 20% of 500 = 100, and 1/4 of 100 = 25. | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | e082c2c89948 | 37 | 250 | [
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1,
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20,
21,
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23,
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23,
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26,
27
] | 0 | \boxed{10} This is a classic problem and the solution is 10. To get the answer, you need to first find the number of four headed clovers. 20% of 500 is 100. 1/4 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 639ff34d55f7 | 37 | 250 | [
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1,
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20,
21,
22,
23,
24,
23,
24,
23,
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25,
26,
27
] | 0 | \boxed{50} Since 25% of 2048 = 512, and 2048 is not a perfect square, but 2048 = 2^11. 2048 is not a perfect square, but 2048 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 078329b5e7f8 | 36 | 250 | [
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12,
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21,
26,
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] | 0 | 2
## Step 1: Calculate the total amount of clo clo Clo Clo
There are 500 total.
## Step 20 = 4 4
The 4 is 20.
## Step 4 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | ae5fab67f1d9 | 37 | 250 | [
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] | 0 | \boxed{5} \boxed{5} \boxed{5} \boxed{5} \boxed{5} \boxed{5} \boxed{5} \boxed{5} \boxed{5} \boxed{5} | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 7ff486462b6c | 37 | 250 | [
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17,
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23,
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25,
26,
27
] | 0 | \boxed{10}
Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem Start
A snail is at the bottom of a 20-meter well. Each day, | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 4c422297c8e1 | 36 | 250 | [
0,
1,
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17,
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23,
24,
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26,
27
] | 0 | \boxed{2}
## Step 1: Calculate the number of four-leaved clovers
First, we need to find the number of four-leaved clovers in the field. Since 20% of the 500 clovers have four | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | e532e5e4e930 | 37 | 250 | [
0,
1,
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20,
21,
22,
23,
24,
25,
26,
27
] | 0 | \boxed{4}
## Step 1: Calculate the number of four-leaved clovers
First, we need to find the number of four-leaved clovers in the field. Since 20% of the 500 clovers have four | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 2228c78d8497 | 37 | 250 | [
0,
1,
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3,
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16,
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22,
23,
24,
25,
26,
27
] | 0 | \numberz
import data structures to count the number of integers in the range from 1 until the first of the two numbers that are square is common common to both the set of numbers that are written in using using the using the base base of | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 63574f481716 | 37 | 250 | [
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27
] | 0 | \boxed{100}
Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem
In a field of 500 clovers, 20 of them have four leaves and one | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 3b6bb23b58f6 | 37 | 250 | [
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1,
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3,
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19,
20,
21,
22,
23,
22,
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22,
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25,
26,
27
] | 0 | \boxed{8}
## Step 1: Calculate the number of four-leaved clovers
To find the number of four-leaved clovers, we need to calculate 20% of 500. This means we need to find 20 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | b24aa82058aa | 37 | 250 | [
0,
1,
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26,
27
] | 0 | \boxed{100]}
Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem Start
What is the value of x in the equation 2x-3x-2 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | e07c7f851317 | 37 | 250 | [
0,
1,
2,
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17,
18,
19,
20,
17,
18,
19,
20,
21,
22,
25,
26,
27
] | 0 | **half of the 1/4 of the 4角 leaves
which is ½ths of an 4角
which is 2
**½ of 2 is
**½
**½
**½
**½
| 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 277474d485f7 | 37 | 250 | [
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23,
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25,
26,
27
] | 0 | was previously 0, now 500/2 - 500/4 = 125/2 = 156.5, round down to the nearest whole number, 156.
Note: The original question was phrased unclear, but | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 06a16df1be37 | 37 | 250 | [
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26,
27
] | 0 | 82
The number of days in December, April, and September are 31, 12, and 12 respectively. If the rest of the, 99 total possible, days in a year except mentioned above are 12, 12 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 625c81bbac99 | 37 | 250 | [
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1,
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26,
27
] | 0 | \boxed{20}
## Step 1: Calculate the number of clovers that are both purple and four-leaved.
To find the exact count of clo clo. we need to calculate 20% of the field's clo. | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | b7ee028f541f | 37 | 250 | [
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26,
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] | 0 | \boxed{40} (Note: This answer is not 4% of 500, but 25% of 20% of 500, which is 2.5% of 500, or 2.5 x .01 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 99bb2054533f | 37 | 250 | [
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21,
22,
27
] | 0 | 125
Swe Swe is going to town meeting to represent his uncle. he has already invited 5 friends and will invite 5 more each time he invites his uncle. inviteation costs $0.05 per person. How much will | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 636fc83e93f3 | 37 | 250 | [
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] | 0 | 0 \ 10 5 \ 10
## 500
## 2
## 5
## 5
## \ 10 5 0 5 3 5
## 5
## 1
| 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | a61007985bdd | 37 | 250 | [
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26,
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] | 0 | [HwGwCoAr]
I'll first for 50% of the entire population, which is 125, because 25% of 10 is quarter of 25, which is quarter of 250, which is | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 211473fa5397 | 37 | 250 | [
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grade-school-math/grade_school_math/data/train.jsonl:1649 | dda895aab529 | 37 | 250 | [
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] | 0 | \boxed-tooljtsn
## Step 1: Calculate the number of four headed clovers
First, we need to find out how many of the 500 total clovers are four headed. To do this, we calculate 20% | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | e77f668d9654 | 37 | 250 | [
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1,
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26,
27
] | 0 | $\boxed{10}$
## Step 1: Calculate the total number of four headed clovers in the field.
First, we need to find out how many of the 500 clovers are four headed. Since 20% of the clo | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 322887795565 | 36 | 250 | [
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] | 0 | \boxed{50}
Solve the 2 math problems and output the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem 1
### Problem End
### Problem 2
### Problem End
### Problem 3 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 21cdf67ca341 | 36 | 250 | [
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25,
26,
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] | 0 | 4
Solve the following math problem and output Only the final answer, formatted strictly as \boxed{Answer}.
### Problem Start
[question] of these is a red | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 675cbd9acbcd | 36 | 250 | [
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27
] | 0 |
First, find the total number of four-leaved and purple flowers:
4.5% of 500 = 20% of 500/2 = 20/2 = 10
Then, 10/ 4 = 2 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | ca0a8ddc753e | 37 | 250 | [
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1,
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26,
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] | 0 | \boxed{40}
## Step 1: Calculate the number of four-leaved clovers
20% of 500 clovers is 0.2 * 500 = 100 clovers.
## Step 2: Calculate the number of | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 70d46d63ab6c | 37 | 250 | [
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1,
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1,
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] | 0 | \boxed{50}plural of "a" that ends in -s. What is the correct answer?
## Step 1: First, let's find 20% of 500, which represents the number of four-leaved clovers | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | fb1029c608cd | 37 | 250 | [
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###Problem Answer
###Problem
###Answer
###ANSWER
###Problem
###ANSWER
###ANSWER
###ANSWER
###ANSWER
###ANSWER
###ANSWER
###ANSWER
###ANS | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | a10f5a27ec51 | 37 | 250 | [
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] | 0 | 500
In 500
In 500
In 500
In 500
In 500
In 500
In 500
In 500
In 500
In 500
In 500
In 500 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | d9999d779e79 | 37 | 250 | [
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] | 0 | 1+2+3+1+4+6+1+1+1+1+2+Zero+1+1+4+1+1+1+1+1+Zero+1+1+4+1 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 52e3cee15405 | 36 | 250 | [
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] | 0 | \boxed{50} | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 847d95d28bad | 37 | 250 | [
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19,
20,
21,
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19,
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23,
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25,
26,
27
] | 0 | \boxed{125}
Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem Set 1: Algebraic Equations and Inequalities
Solve the following system of inequalities | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 702aa2620a31 | 36 | 250 | [
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26,
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] | 0 | 0
and 3, v. I
: a ...ago... w 1
0
5 0 0 0
0 ippers
| 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | a61200130640 | 37 | 250 | [
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] | 0 | \boxed{2}
## Step thought process process
steps are included in the reasoning thought process process for math math math math math a is a number number number 1 1 1 1 1 1 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | cf1e9e981ff8 | 37 | 250 | [
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18,
19,
20,
21,
22,
25,
26,
27
] | 0 | \boxed{20}
The final answer is: \boxed{20} \u{u}\u{u}} \u{u}} \u{u}} \u{u}} \u{u}} \u{u}} \ | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | f35c88a08bf8 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
8,
8,
9,
10,
11,
12,
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15,
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18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | \boxed{125}
## Step 1: Calculate the number of four-leaved clovers
First, we need to find out how many clovers have four leaves. Since 20% of the 500 clovers have four leaves, we | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 9612409171a1 | 37 | 250 | [
0,
1,
2,
3,
4,
7,
8,
8,
8,
9,
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13,
14,
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18,
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20,
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23,
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] | 0 | 2
### Problem Start
[question] | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 64110a2e1260 | 36 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
8,
8,
9,
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11,
12,
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14,
15,
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17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | 500 x (chet: 500 + (chet: \frac \ Clover 4 - ( 4)Q
=chet4 ceph ceph 4 4 ceph ceph | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 540e6334d1c3 | 37 | 250 | [
0,
1,
5,
6,
7,
8,
8,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | 4
## Step 1: Understand the problem
The problem is asking for a specific condition, and the answer is a specific number. The answer is 4.
## Step 2: Realize the answer is not just a simple math | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 193d912e8244 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
6,
6,
6,
6,
7,
8,
8,
8,
9,
10,
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17,
18,
19,
20,
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23,
24,
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] | 0 | 12
- of aft, 15., (1, a):ждvis ( 2, 2 2 3 4 5 6 7 8 9 0 1 2 | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 464e0af237ab | 37 | 250 | [
0,
1,
2,
3,
2,
3,
2,
3,
4,
5,
6,
7,
8,
8,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | \boxed{100} ```
## Step 1: Calculate the number of four-leaved clovers
First, we need to find out how many clovers have four leaves. We know that 20% of the total number of clo | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 64e27ea2276a | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
21,
22,
23,
24,
25,
26,
27
] | 0 | \boxed{125}
The final answer is $\boxed{125}$.
Explanation: 500 x 0.25 = 125.
The final answer is $\boxed{125}$.
The final answer is $\boxed{125} | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 32a363e04f51 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
7,
8,
9,
10,
11,
12,
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18,
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20,
21,
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23,
24,
25,
26,
23,
24,
25,
26,
27
] | 0 | \boxed{22}
## Step 1: First, let's find out how many four-leaved clovers there are in the field.
To find the number of four-leaved clovers, we need to calculate 20% of | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 146f0730b9bf | 36 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
15,
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17,
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20,
21,
22,
23,
20,
21,
22,
23,
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25,
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27
] | 0 | "see Aug 20 and Sep Sep Nov 10 for a solution and a graph that illustrates the same idea. to the left, we have the following table of values for the two "hacking" operations ++ and ×-+ ( + | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 8f647171c758 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
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25,
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] | 0 | \boxed{10}
Solve the following math problem and output ONLY the final answer directly, formatted strictly as \boxed{ANSWER}.
### Problem Start
A rectangular garden measures 10 meters by 5 meters. A path that is | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 45cae4dc8640 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
10,
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21,
22,
23,
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] | 0 | \boxed{20} is the number of purple clovers in the field.
## Step 1: Identify the problem
The problem is to find the number of purple clovers in a field of 500, where 20% of the total | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | d5035622c97b | 36 | 250 | [
0,
1,
2,
3,
4,
5,
5,
5,
5,
5,
6,
7,
8,
9,
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] | 0 | 20
## Step 1: Step 1: Determine the number of leaves on the plant
To solve the problem, we need to first determine the number of leaves on the plant. Since there are 4 leaves on the plant, we | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 52d88e7b0060 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
8,
9,
10,
11,
8,
9,
10,
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24,
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] | 0 | \boxed{10}
### Problem Initiative
To solve this problem, we need to break it down into steps.
### Step 1: Calculate the number of clovers with four leaves
First, we need to find the number of clovers with | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 3d04bccd5537 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
11,
11,
11,
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19,
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24,
25,
25,
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27
] | 0 | \boxed{4}
In a field of 500, the number of four-leaved clovers is 20.
The number of four-leaved clovers is 20.
The number of four-leaved clovers is 20.
| 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | f6335fc47bc3 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
8,
8,
8,
8,
9,
10,
11,
12,
13,
14,
15,
16,
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18,
19,
20,
21,
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23,
24,
25,
25,
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27
] | 0 | 1 directly, but the areFTA.swingittle1ichoFTAichoщиvetFTAicho stake stakeicholegationicholegationicholegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegationlegation | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | f5b9b08f7fcb | 37 | 250 | [
0,
1,
2,
3,
4,
1,
2,
3,
4,
1,
2,
3,
4,
5,
6,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
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] | 0 | \boxed{200}.
## Step 1: First, we need to identify the total number of purple clovers in the field.
## Step 2: We know that 1/4 of the field is purple clovers, | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | df822f25a785 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
17,
18,
19,
17,
18,
19,
17,
18,
19,
20,
21,
22,
23,
24,
25,
25,
26,
27
] | 0 | 10.6666666667
To, FIFteen, add a two, Make, a, colon, and, a space, and, a hy colon, and, a space, and, a colon colon, and, a space | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 8c670d25bd7c | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
6,
7,
8,
6,
7,
8,
6,
7,
8,
6,
7,
8,
9,
10,
11,
12,
13,
15,
16,
17,
18,
19,
20,
21,
26,
27
] | 0 | 50
## Problem 1: 1. the problem asks for the amount of the total number of the clovetier's house, but the house is not a house, the answer is 0
## Problem 2 the problem asks | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 20dcc0213a37 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
7,
8,
7,
8,
7,
8,
7,
8,
9,
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11,
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13,
14,
15,
16,
17,
18,
19,
20,
21,
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27
] | 0 | $\boxed{2}$
## Step 1: Calculate the number of four-leaved clovers in the field
We know that 20% of the clovers have four leaves, and we can set up the equation $\ (500 \ ( | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 17b5dbac8a9e | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
7,
8,
7,
8,
7,
8,
7,
8,
9,
10,
11,
12,
13,
14,
15,
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17,
18,
19,
17,
18,
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17,
18,
19,
17,
18,
19,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | 0 is not allowed in the division, so 0 is not the answer, 0. The correct answer is 120.
Answer: 0 is not a number, 3 of 3 of | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 8acbf7b7f282 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
10,
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19,
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21,
21,
21,
21,
22,
23,
24,
25,
25,
26,
27
] | 0 | 25
## Step 1: Understand the problem
The problem is asking to find the number of clovers that are both four-leaf and four-leaf clovers.
## Step 2: Calculate the number of four-leaf clovers | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 288b2daf6621 | 36 | 250 | [
0,
1,
2,
3,
4,
4,
4,
5,
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] | 0 | sized ={
\section source = Source
title title source
Daviaoh Frag Lorem Lorem恒re. Lorem Loremasures, Lorem恒. Lorem Lorem Frag Herm Lorem Lorem Frag Lorem Lorem Frag Lorem Loremimals. Lorem Lorem Frag Herm Herm. | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | ecc0596e37b5 | 37 | 250 | [
0,
1,
5,
6,
7,
8,
8,
8,
9,
10,
11,
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13,
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12,
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12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
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] | 0 | 50% of the time, 20% of the area is not enough to be considered a safe space. The area is not safe, but it is not purple either. This is a classic example of how to make a decision that is both | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 3c9e208c9341 | 36 | 250 | [
0,
1,
2,
3,
4,
5,
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7,
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17,
18,
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17,
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16,
17,
18,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | (60/100) (20 of 100 = 4, so 4 of 10 = 2, which is correct, now you have 2 groups of 8 each, so you can divide 2 by 2 to get | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 7797dd0399e1 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
6,
7,
8,
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9,
6,
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8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
18,
19,
20,
21,
18,
19,
20,
21,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | \boxed{9}
Note: The final answer should be followed by a blank line that ends with a corresponding closing bracket.
The same format instructions apply to the problem and any subsequent "Problem" and "Answer" sections.
The same format | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | b44440ebbcff | 36 | 250 | [
0,
1,
2,
3,
3,
3,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
10,
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12,
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10,
11,
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13,
10,
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14,
15,
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17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27
] | 0 | \boxed{4}
Note: The final answer is not a real solution, but a placeholder for the format of the answer. The actual solution is not provided as the problem does not require a numerical value to solve the problem. The answer is | 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | 2da385b02fb6 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
12,
12,
12,
13,
15,
16,
17,
18,
19,
20,
21,
22,
23,
22,
23,
22,
23,
24,
25,
26,
27
] | 0 | 88
S:
d = size of each of the 4 parts
### ( ' =
1 part = 0.5 = 5/ = 3/
= 5/5 = = 2
| 1 |
grade-school-math/grade_school_math/data/train.jsonl:1649 | afc207904f67 | 37 | 250 | [
0,
1,
2,
3,
4,
5,
6,
7,
8,
6,
7,
8,
9,
10,
11,
12,
13,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
22,
23,
24,
25,
26,
27
] | 0 | \boxed{20}
## Step 1: First, we need to find the number of four-leaved clovers in the field.
To find the number of four-leaved clovers, we need to calculate 20% of 500.
| 1 |
End of preview. Expand in Data Studio
RE-PoLar: every program-of-layers our searches evaluated
- Authors: Justus Westerhoff*, Stephan Olbrich*, Hatem Oraby, Walter Senn, Matthew E. Larkum, Felix Alexander Gers
- *Shared first authorship; order chosen at random.
- Affiliations: Berliner Hochschule für Technik (BHT), Humboldt-Universität zu Berlin, University of Bern
- Project page (interactive search-tree explorer): https://datexis.github.io/RE-PoLar/
- Code: https://github.com/DATEXIS/RE-PoLar
- Paper: Programs-of-Layers in LLMs through the Lens of Cortical Areas
- Reproduced work: PoLar (Li et al., ICML 2026)
Overview
A program-of-layers runs a transformer's layers in a different order than the standard forward pass: some layers are skipped, some segments repeated. PoLar finds such programs per question with a Monte Carlo Tree Search (MCTS) and trains a router that predicts one from the question. This dataset holds everything our reproduction's searches, routers and fixed programs produced:
- Main track (our DART-Math split, our search,
oneshotprompt): 6 models × 3 search seeds × 5 difficulty levels × 2,000 questions. Every program each search evaluated, with its verdict (56.6M programs in the whole dataset, 4.6M of them solving their question), the model's generated answer behind each verdict (53.2M answers), the trained routers and their test results, the best fixed program per level, and the menu cross-execution. - PoLar split (
polar_split/, paper Appendix B): Li et al.'s released search on their own MATH question sets, 4 models × 2 prompts (polar_default,oneshot) × 3 seeds, with 4-fold cross-validated routers and fixed programs, and routers evaluated out of distribution (ASDiv, MAWPS, MMLU-Pro). - Cross-checks (
cross_checks/): their search on our split and our search on theirs.
Models: Qwen1.5-MoE-A2.7B-Chat, Qwen2.5-3B-Instruct, Qwen2.5-7B-Instruct, Qwen3-8B, Qwen3-32B, LLaMA-3.2-3B-Instruct.
Explore the search trees
The dataset viewer above shows the searches as tables: pick programs_<model> for the
programs and verdicts of each question (split = search seed), answers_<model> for the
generated answers, polar_split_programs_<model> for the PoLar split. The search trees
themselves, node by node, are on the
project page's interactive explorer.
Layout
mcts_results/<model>/seed<S>/dart-math-diff-<L>/merged_mcts_samples.json.gz programs + verdicts per question
generated_answers/<model>/seed<S>/diff<L>.jsonl.gz one generated answer per evaluation
router_checkpoints/<model>/seed<S>/diff<L>.pt trained routers (weights only)
router_results/<model>/seed<S>/diff<L>.json router test results, per question
fixed_program/<model>/seed<S>/diff<L>.json fixed-program baseline, per question
menu_crossexec/<model>/diff<L>.json.gz top-100 programs x 2,000 questions
menu_crossexec_heldout/<model>/diff<L>.json.gz top-100 by train+val, run on the 500 test questions
polar_split/... same artifacts under <model>/<prompt>/seed<S>/
cross_checks/{their_search_our_split,our_search_polar_split}/...
DATA.md full schema of every file
<model>: qwen15_moe_a27b, qwen25_3b, qwen25_7b, qwen3_8b, qwen3_32b, llama32_3b;
<S>: search seed 42, 1, 7; <L>: difficulty level 1-5. DATA.md documents every
field, the search configurations and which runs have no answer texts.
Download and use
The code expects the files under data/ of a clone of the repository:
git clone https://github.com/DATEXIS/RE-PoLar && cd RE-PoLar
pip install -U huggingface_hub
hf download SWAMBIT/re-polar --repo-type dataset --local-dir data # everything, ~6 GB
hf download SWAMBIT/re-polar --repo-type dataset --local-dir data --include "router_results/*" "fixed_program/*"
One file with the datasets library, or directly:
from datasets import load_dataset
programs = load_dataset("SWAMBIT/re-polar", "programs_qwen3_8b", split="seed42")
print(programs[0]["sample_info"]["query_id"], len(programs[0]["final_valid_transitions"]))
import gzip, json
router = json.load(open("data/router_results/qwen3_8b/seed42/diff1.json"))
print(router["overall"]) # Base tau=0, router p@1 and p@5 on the 500 test questions
with gzip.open("data/generated_answers/qwen3_8b/seed42/diff1.jsonl.gz", "rt") as f:
print(json.loads(next(f)))
Questions are not included
No file holds question text or reference answers, only query_ids. The MATH half of the
DART-Math pool traces back to the delisted hendrycks/competition_math, so the questions are
rebuilt locally from the public pool with the repository's code and joined on query_id
(re_polar.datasets.dart_math, re_polar.datasets.polar_split,
re_polar.datasets.attach_mcts_questions; see the repository README). The models sometimes
restate their question in a generated answer; such restatements are replaced by
[question] (DATA.md, "Generated text").
Citation
@article{westerhoff2026repolar,
title={Programs-of-Layers in LLMs through the Lens of Cortical Areas},
author={Westerhoff, Justus and Olbrich, Stephan and Oraby, Hatem and Senn, Walter and Larkum, Matthew Evan and Gers, Felix Alexander},
journal={arXiv preprint arXiv:2609.31360},
year={2026}
}
License
CC BY-NC 4.0. The generated answers and verdicts are outputs of the models above and also fall under their licenses: Qwen1.5-MoE-A2.7B-Chat (Tongyi Qianwen License), Qwen2.5-3B-Instruct (Qwen Research License), Qwen2.5-7B-Instruct, Qwen3-8B and Qwen3-32B (Apache 2.0), LLaMA-3.2-3B-Instruct (Llama 3.2 Community License; Built with Llama). The routers embed questions with Qwen3-Embedding-0.6B (Apache 2.0). The code is MIT-licensed, see the repository.
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