Rabdos Math Bench

74 problems · Updated on August 5, 2026

For research-level mathematical reasoning.

RankModel

Pass rate

Leaderboard, ranked by average score.
Rank	Provider	Model	Reasoning effort	Pass rate
1	OpenAI	GPT-5.6 Sol	max	40.5%
2	Anthropic	Claude Opus 5	max	37.2%
3	Anthropic	Claude Fable 5	max	32.4%
4	Moonshot AI	Kimi K3	max	17.6%
5	xAI	Grok 4.5	xhigh	16.2%
6	Meta	Muse Spark 1.1	xhigh	14.9%
6	Alibaba	Qwen 3.8 Max	max	14.9%
8	Google	Gemini 3.6 Flash	high	13.5%
9	Z.ai	GLM 5.2	xhigh	10.8%
9	DeepSeek	DeepSeek V4 Flash	max	10.8%
11	NVIDIA	Nemotron 3 Ultra	high	9.5%

Sample problems

Shadows of uniform set families

Extremal set theory

For a finite family $\mathcal{F}$ of distinct $k$ -element sets, let $d(\mathcal{F})$ be the collection of all $(k-1)$ -element sets contained in at least one member of $\mathcal{F}$ , and write $D(\mathcal{F}) = |d(\mathcal{F})|$ . Let $M(k)$ be the largest positive integer that is not equal to $D(\mathcal{F})$ for any finite family $\mathcal{F}$ of distinct $k$ -element sets. For fixed $k$ and $t$ , let $L_k(t)$ be the smallest integer $L$ such that every integer from $L$ through $tk$ is equal to $D(\mathcal{F})$ for some $t$ -member family $\mathcal{F}$ of distinct $k$ -element sets. Let $\tau(k)$ be the largest $t$ in $\{1, \ldots, k+1\}$ with $L_k(t) \ge tk - k + 2$ .

Let $h(k)$ be the least integer $j \ge 0$ such that

\binom{j+3}{2} \ge \tau(k).

Among the integers $k$ with $374 \le k \le 25000$ for which

\gcd\!\left(M(k),\, \tau(k)\,(h(k)+1)\right) = 1,

what is the sum of those $k$ ?

Low-stem squares in the cobar complex

Algebraic topology

Work over $\mathbb F_2$ . Let $A=\mathbb F_2[\xi_1,\xi_2,\ldots]$ , with $|\xi_i|=2^i-1$ and coproduct

\Delta(\xi_n)=\sum_{i=0}^{n}\xi_{n-i}^{\,2^i}\otimes\xi_i,

where $\xi_0=1$ , extended multiplicatively. Let $\epsilon\colon A\to\mathbb F_2$ be the augmentation determined by $\epsilon(\xi_i)=0$ , and let $\overline A=\ker\epsilon$ . For $s\ge1$ let $C^s=\overline A^{\otimes s}$ , and let $C^0=\mathbb F_2$ . Write a basis element of $C^s$ as $[a_1|\cdots|a_s]$ , where each $a_i$ is a positive-degree monomial in $A$ , and give this word stem $\sum_{i=1}^{s}|a_i|-s$ .

For $a\in\overline A$ , write its reduced coproduct as

\overline\Delta(a)=\Delta(a)-a\otimes1-1\otimes a=\sum a'\otimes a''.

Define

d[a_1|\cdots|a_s]=\sum_{j=1}^{s}\ \sum\,[a_1|\cdots|a_j'|a_j''|\cdots|a_s],

where the inner sum runs over the terms of $\overline\Delta(a_j)$ , and extend $d$ linearly. Multiply words by concatenation. Let $\overline H=H^\ast(C,d)\big/\left\langle[\xi_1],[\xi_1^2]\right\rangle$ , where the denominator is the ideal generated by the two indicated cohomology classes. The product on cohomology induced by concatenation gives a commutative product on $\overline H$ . Since the ideal is bihomogeneous, the gradings by word length and internal degree, and hence the decomposition by stem, descend to $\overline H$ .

For $x\in\overline H$ , define its retained square $q(x)$ to be the sum of the homogeneous components of $x^2$ having stem at most $13$ . Let $V$ be the $\mathbb F_2$ -subspace of $\overline H$ spanned by the classes represented by

[\xi_1^4],\qquad [\xi_1^4|\xi_1^4],\qquad [\xi_1^4|\xi_1^4|\xi_1^4],\qquad [\xi_1^8].

How many ordered nine-tuples $(x_1,\ldots,x_9)\in V^9$ simultaneously satisfy $x_1+\cdots+x_9=0$ , have exactly four entries $x_i$ for which $q(x_i)\ne0$ , and span $V$ ?

Crossings of polygonal curves

Discrete geometry

A closed polygonal curve in the plane is called generic if its vertices are distinct, no three vertices are collinear, no vertex lies in the relative interior of a nonincident edge, and no three edges have a common point. Thus every intersection between nonadjacent edges is a transverse double crossing.

A generic closed polygonal curve $X$ has exactly two inflection edges and exactly $95$ double supporting lines in total. An edge is an inflection edge if the relative interiors of its two neighboring edges lie in opposite open half-planes bounded by the line containing it. A line through two nonadjacent vertices is double supporting if, at each of the two vertices, the relative interiors of both incident edges lie in the same open half-plane bounded by the line. It is exterior if the two corresponding half-planes are the same, and interior if they are opposite.

Let $C$ be the set of positive crossing counts attained by all generic closed polygonal curves having no inflection edges and exactly $95$ double supporting lines in total, where a crossing is an intersection between the relative interiors of two nonadjacent edges. Form every nonnegative integer that can be written as a finite sum, with repetitions allowed, of members of $C$ , including the empty sum. Only finitely many nonnegative integers are missing, and the largest missing integer is one less than the excess of the number of exterior double supporting lines of $X$ over the number of interior double supporting lines of $X$ .

How many crossings does $X$ have?

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