Datasets:
task_id stringlengths 6 6 | category stringclasses 6
values | prompt stringlengths 96 245 | api_description stringclasses 1
value | expected_output stringlengths 19 97 |
|---|---|---|---|---|
cm-000 | recurrences | The recurrence T(0) = 0, T(n) = 2*T(n-1) + 1 counts the moves to solve the Tower of Hanoi. Give T(10), T(20) and T(64). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[1023, 1048575, 18446744073709551615]]} |
cm-001 | recurrences | The recurrence L(0) = 1, L(n) = L(n-1) + n counts the regions n straight lines cut the plane into. Give L(5), L(50) and L(1000). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[16, 1276, 500501]]} |
cm-002 | recurrences | Bent lines: Z(0) = 1, Z(n) = Z(n-1) + 4*n - 3 counts the regions n V-shaped bent lines cut the plane into. Give Z(4), Z(30) and Z(500). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[29, 1771, 499501]]} |
cm-003 | recurrences | In the Josephus problem, n people stand in a circle and every SECOND person is eliminated, starting by eliminating person 2. Give the 1-indexed position of the survivor for n = 10, n = 100 and n = 1000. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[5, 73, 977]]} |
cm-004 | recurrences | The recurrence f(0) = 1, f(1) = 1, f(n) = f(n-1) + f(n-2) is the Fibonacci sequence. Give f(30), f(60) and the number of DECIMAL DIGITS of f(1000). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[1346269, 2504730781961, 209]]} |
cm-005 | recurrences | A recurrence a(0) = 2, a(n) = 3*a(n-1) - 1. Give a(5), a(15) and a(40). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[365, 21523361, 18236498188585393202]]} |
cm-006 | recurrences | The number of ways to tile a 2-by-n strip with 1-by-2 dominoes satisfies D(0) = 1, D(1) = 1, D(n) = D(n-1) + D(n-2). Give D(12), D(25) and the smallest n for which D(n) exceeds one million. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[233, 121393, 30]]} |
cm-007 | sums | Compute the sum of k**3 for k = 1 to n, for n = 10, n = 100 and n = 1000. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[3025, 25502500, 250500250000]]} |
cm-008 | sums | Compute the sum of k * 2**k for k = 0 to n, for n = 10, n = 20 and n = 50. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[18434, 39845890, 110338190870577154]]} |
cm-009 | sums | The harmonic number H_n is the sum of 1/k for k = 1 to n, computed EXACTLY as a rational. Give H_10, H_20 and H_50, each as [numerator, denominator] in lowest terms. Return three rows, in that order. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[7381, 2520], [55835135, 15519504], [13943237577224054960759, 3099044504245996706400]]} |
cm-010 | sums | Compute the telescoping sum of 1/(k*(k+1)) for k = 1 to n exactly, for n = 10, n = 100 and n = 1000. Give each as [numerator, denominator]. Return three rows. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[10, 11], [100, 101], [1000, 1001]]} |
cm-011 | sums | Compute the alternating sum of (-1)**(k+1) * k for k = 1 to n, for n = 9, n = 10 and n = 101. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[5, -5, 51]]} |
cm-012 | sums | Compute the sum of 1/(k**2) for k = 1 to n EXACTLY as a rational, for n = 6 and n = 12, and give the numerator of the n = 6 value alone as a third answer. Return three rows: [num, den] for n=6, [num, den] for n=12, and [numerator] again for n=6. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[5369, 3600], [240505109, 153679680], [5369]]} |
cm-013 | sums | Compute the sum of k * (k + 1) for k = 1 to n, for n = 10, n = 100 and n = 5000. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[440, 343400, 41691670000]]} |
cm-014 | integer-functions | Compute the sum of floor(sqrt(k)) for k = 1 to n, for n = 100, n = 1000 and n = 10000. Use exact integer square roots, not floating point. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[625, 20615, 661750]]} |
cm-015 | integer-functions | Compute the sum of floor(n / k) for k = 1 to n, for n = 20, n = 200 and n = 2000. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[66, 1098, 15518]]} |
cm-016 | integer-functions | For x = 17/5 as an exact rational, give floor(x), ceil(x), floor(-x) and ceil(-x). Return one row of the four integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[3, 4, -4, -3]]} |
cm-017 | integer-functions | Count the decimal digits of 2**n for n = 10, n = 100 and n = 1000, computing the powers exactly rather than with logarithms. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[4, 31, 302]]} |
cm-018 | integer-functions | How many integers k in 1..n satisfy floor(n/k) == 1? Give the count for n = 10, n = 100 and n = 1000. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[5, 50, 500]]} |
cm-019 | integer-functions | The Spectrum of 3/2 is the sequence floor(k * 3/2) for k = 1, 2, 3, .... Give its first eight terms, computed with exact rationals. Return one row of the eight integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[1, 3, 4, 6, 7, 9, 10, 12]]} |
cm-020 | binomials | Give C(20, 10), C(52, 5) and C(100, 50) as exact integers. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[184756, 2598960, 100891344545564193334812497256]]} |
cm-021 | binomials | Compute the sum of C(n, k)**2 for k = 0 to n, for n = 5, n = 10 and n = 20. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[252, 184756, 137846528820]]} |
cm-022 | binomials | The hockey-stick identity sums C(k, r) for k = r to n. Compute it for (r, n) = (3, 10), (5, 20) and (2, 100). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[330, 54264, 166650]]} |
cm-023 | binomials | Compute the alternating sum of (-1)**k * C(n, k) for k = 0 to n, for n = 0, n = 1 and n = 7. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[1, 0, 0]]} |
cm-024 | binomials | Vandermonde's convolution sums C(m, k) * C(n, r - k) over k. Compute it for (m, n, r) = (5, 7, 4), (10, 10, 10) and (8, 3, 5). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[495, 184756, 462]]} |
cm-025 | binomials | Give the Catalan numbers C(2n, n) // (n + 1) for n = 5, n = 10 and n = 20. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[42, 16796, 6564120420]]} |
cm-026 | special-numbers | The Stirling numbers of the SECOND kind S(n, k) count partitions of n labelled items into k non-empty blocks, with S(0,0)=1 and S(n,k) = k*S(n-1,k) + S(n-1,k-1). Give S(6,3), S(10,5) and S(12,4). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[90, 42525, 611501]]} |
cm-027 | special-numbers | The Bell number B(n) is the total number of partitions of n labelled items, the sum of S(n, k) over all k. Give B(5), B(8) and B(12). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[52, 4140, 4213597]]} |
cm-028 | special-numbers | The unsigned Stirling numbers of the FIRST kind c(n, k) count permutations of n items with k cycles, with c(0,0)=1 and c(n,k) = (n-1)*c(n-1,k) + c(n-1,k-1). Give c(5,2), c(8,3) and c(10,4). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[50, 13132, 723680]]} |
cm-029 | special-numbers | The Eulerian number A(n, k) counts permutations of n items with exactly k ascents, with A(n,k) = (k+1)*A(n-1,k) + (n-k)*A(n-1,k-1) and A(0,0)=1. Give A(5,2), A(7,3) and A(8,4). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[66, 2416, 15619]]} |
cm-030 | special-numbers | Give the exact rational value of the harmonic-like sum of 1/(2k-1) for k = 1 to n, for n = 5 and n = 12, each as [numerator, denominator]. Return two rows. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[563, 315], [744355888, 334639305]]} |
cm-031 | special-numbers | The double factorial n!! multiplies every other integer down from n. Give 10!!, 11!! and 15!! as exact integers. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[3840, 10395, 2027025]]} |
cm-032 | number-theory | Give gcd(1071, 462), and the NUMBER of division steps Euclid's algorithm takes on that pair, counting each replacement of (a, b) by (b, a mod b) as one step until b becomes 0. Return one row [gcd, steps]. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[21, 3]]} |
cm-033 | number-theory | Euler's totient phi(n) counts the integers in 1..n coprime to n. Give phi(36), phi(97) and phi(1000). Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[12, 96, 400]]} |
cm-034 | number-theory | Fibonacci numbers satisfy gcd(F(m), F(n)) = F(gcd(m, n)) with F(1)=F(2)=1. Give gcd(F(12), F(18)), gcd(F(15), F(25)) and gcd(F(20), F(30)) as exact integers. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[8, 5, 55]]} |
cm-035 | number-theory | Give the exponent of the prime 2 and of the prime 5 in the factorisation of 100!, and the number of trailing zeros of 100! in decimal. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[97, 24, 24]]} |
cm-036 | number-theory | Give the number of primes below 100, below 1000 and below 10000, counted with a sieve. Return one row of the three integers. | You are solving discrete-mathematics problems EXACTLY, by computing them
in Python rather than recalling a formula's value.
Available imports: math, fractions (Fraction), itertools. Nothing else is needed.
ANSWERS MUST BE EXACT. There are no floats in this environment:
* an integer answer is a plain Python int (the... | {"rows": [[25, 168, 1229]]} |
concretemath-tasks-v1
37 exact discrete-mathematics tasks for the
concretemath-env
RL environment, on the topics of Concrete Mathematics (Graham, Knuth & Patashnik):
recurrences, sums, integer functions, binomial coefficients, special numbers and elementary
number theory.
| field | meaning |
|---|---|
task_id |
cm-000 … cm-036 |
category |
recurrences / sums / integer-functions / binomials / special-numbers / number-theory |
prompt |
the question and the exact shape of the answer |
api_description |
the shared preamble (exact-arithmetic rules, the [num, den] convention) |
expected_output |
JSON {"rows": [...]}, computed by executing a reference solution |
Categories: recurrences 7, sums 7, integer-functions 6, binomials 6, special-numbers 6,
number-theory 5. No dependencies — math, fractions and itertools are all standard library.
Original problems, not the book's exercises
Nothing is copied. Every task is written fresh on the topics the book teaches — a copyright decision and the stronger evaluation one, since the book's exercises and their answers are all over the public web and can be recalled rather than derived.
Not a single float
Integers stay integers (Python's are unbounded) and rationals are returned as
[numerator, denominator] in lowest terms. That is the subject rather than a style rule:
H_10 is 7381/2520, not 2.9289682539682538. sum(1/k ...) is a wrong answer to a
harmonic-number question and sum(Fraction(1, k) ...) is a right one, so the grader is built
to tell those apart — a float never passes for an integer, even when equal in value.
Answer keys are executed, and cross-checked against a second method
Where a closed form exists, the task computes the answer by ITERATION and asserts the closed form agrees — so the key is never just "whatever the first implementation printed". Those assertions are inside the builder, so the dataset fails to build if the two ever disagree.
That earned its keep immediately: the closed form written for a(n) = 3·a(n-1) - 1 was
(3^n + 1)/2, which is the answer to a different recurrence. The iteration was right and the
check was wrong — exactly the failure a cross-check exists to catch, and it would otherwise
have shipped as a wrong answer key.
Verify with python environments/concretemath_env/build_tasks.py --verify (37/37).
Source: https://github.com/eltociear/my-molt-agent/tree/main/environments/concretemath_env
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