💬 NLP

MathDuels: Evaluating LLMs as Problem Posers and Solvers

The paper introduces MathDuels, a dynamic self-play benchmark where frontier language models simultaneously act as adversarial problem authors and solvers, utilizing a Rasch model to reveal decoupled capabilities and prevent evaluation saturation through a co-evolving difficulty curve.

Original authors: Zhiqiu Xu, Shibo Jin, Shreya Arya, Mayur Naik

Published 2026-04-24

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CC BY 4.0

Original authors: Zhiqiu Xu, Shibo Jin, Shreya Arya, Mayur Naik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out who the best chess player in the world is.

For a long time, we've done this by giving them a fixed list of 100 puzzles. If they solve 99 of them, they get a high score. But here's the problem: the puzzles are getting too easy. The top players solve them all perfectly, so we can't tell who is actually better. It's like trying to measure the height of giants using a ruler that only goes up to 6 feet; they all just hit the top and look the same.

This paper introduces a new way to test AI math brains called MathDuels. Instead of a static test, it turns math into a living, breathing arena where the AI models play two roles at once: The Architect and The Solver.

Here is how it works, using a simple analogy:

1. The Arena: A "Math Duel"

Imagine a tournament where every participant has to do two things:

Role A: The Architect (Problem Creator): They must design a tricky math puzzle for everyone else to solve.
Role B: The Solver (Problem Solver): They must try to solve the puzzles created by everyone else.

This is inspired by a real historical event from 1535. Two mathematicians, Tartaglia and Fior, didn't just take a test; they challenged each other. They wrote 30 problems for the other to solve. The winner wasn't just the one who could solve the best; it was the one who could create problems the other couldn't crack.

2. The Twist: "Self-Play"

In this AI tournament, the models play against each other in a giant round-robin.

Model A writes 30 hard problems.
Model B, C, D... try to solve Model A's problems.
Then, Model B writes 30 problems, and everyone else tries to solve them.

The system uses a special "referee" (an independent AI) to check if the problems are fair and if the answers are actually correct. If a problem is broken or confusing, it gets thrown out.

3. The Scoreboard: Two Different Skills

The most surprising discovery of this paper is that being good at solving math doesn't mean you are good at making math.

Think of it like a video game:

Some players are Speedrunners: They can finish any level incredibly fast (Great Solvers).
Some players are Level Designers: They can build levels so tricky that even the speedrunners get stuck (Great Architects).

In the past, we only measured the Speedrunners. MathDuels measures both.

The Result: One AI model (Gemini-3.1-Pro-high) wasn't the fastest solver, but it was the best architect. It built puzzles so clever that even the fastest solvers got stuck. Because of this, it ranked #1 overall.
Another model (GPT-5.4-high) was the fastest solver but a mediocre architect. It ranked #2.

4. Why This Matters: The "Moving Target"

In old tests, once the AI gets too smart, the test becomes useless (it hits a "ceiling").
In MathDuels, the difficulty grows with the players.

When a super-smart new AI enters the arena, it doesn't just solve the old puzzles; it creates new, harder puzzles that break the previous champions.
It's like a video game where every time you beat the final boss, the boss upgrades itself to be even harder for the next round. The test never stops being challenging.

5. The "Trap" of Easy Problems

The researchers found that some models try to cheat by writing "trick" questions that look hard but are actually broken or unsolvable. The system has a safety net (the Verifier) that catches these. If a model writes a problem that is mathematically nonsense, it gets penalized. This forces the models to be honest and rigorous, not just tricky.

Summary

MathDuels is a new way to rank AI math skills by making them teach and test each other.

It realizes that creating a hard problem is a different skill than solving one.
It prevents the test from becoming "too easy" because the AI models keep inventing harder challenges for each other.
It gives us a leaderboard that actually changes and evolves, showing us who is truly the "math genius" of the group, not just the one who memorized the answers.

It's the difference between a student who can ace a multiple-choice test and a professor who can design a test that stumps the entire class. MathDuels finds the professors.

1. Problem Statement

Current mathematical benchmarks for Large Language Models (LLMs) are facing a saturation crisis. Static datasets (e.g., MATH, GSM8K) and even live competition problems (e.g., AIME) are being solved with near-ceiling performance by frontier models. This renders them ineffective for differentiating between state-of-the-art systems. Furthermore, the supply of human-authored, research-level problems that are difficult enough to challenge these models is insufficient to keep pace with the rapid advancement of model capabilities. The core challenge is to create a dynamic evaluation framework that prevents performance saturation and can distinguish subtle differences in mathematical competence.

2. Methodology: The MathDuels Protocol

The authors introduce MathDuels, a self-play benchmark where models occupy two simultaneous roles: Problem Author and Problem Solver. The evaluation is fully automated and proceeds in four phases:

A. Arena Setup

Participants: A set of $N$ frontier models.
Task: Each model generates $K$ problems (e.g., 30) across six mathematical domains (Analysis, Algebra, Geometry, Discrete Math, Probability, Applied Math).
Interaction: Every model attempts to solve every problem generated by other models (excluding its own).

B. Three-Stage Generation Pipeline

To ensure high-quality, difficult problems, the authors employ a specific generation process:

Meta-prompting: The model is first asked to generate a prompt for creating a difficult problem within a specific domain, forcing it to reason about difficulty strategies before writing the problem.
Problem Generation: The model uses the generated meta-prompt to create a problem statement ( $q$ ) and a gold answer ( $g$ ).
Difficulty Amplification: The model produces variants of the problem to elicit deeper reasoning, increasing the difficulty.

C. Solving and Verification

Solving: Non-author models attempt to solve the problems, providing a final answer and a reasoning trace.
Symbolic Verification: Answers are checked for symbolic equivalence against the gold answer.
Adversarial Verification: If a solver fails or answers disagree, a dedicated Verifier Model (e.g., GPT-5.4) reviews the problem statement, the gold answer, and solver traces.
- It checks if the problem is well-posed (has a unique solution).
- It selects the correct answer from candidate traces if the author's gold answer was incorrect.
- Ill-posed or ambiguous problems are excluded from scoring.

D. Ranking via Rasch Model

The results are processed using the Rasch model (a form of Item Response Theory) to jointly estimate:

Solver Ability ( $s_m$ ): The probability of a model solving a problem depends on the difference between its ability and the problem's difficulty.
Problem Difficulty ( $d_p$ ): Derived from the success rates of various solvers.
Author Quality ( $R_{author}$ ): Calculated as the mean difficulty of the problems authored by a specific model.
Composite Rating: A weighted average of the Solver Rating and Author Rating ( $R_m = 0.5 R_{solve} + 0.5 R_{author}$ ).

3. Key Contributions

Dual-Role Evaluation Protocol: A novel self-play framework where models are scored on both constructing and solving problems, preventing the "ceiling effect" of static benchmarks.
Decoupling of Capabilities: The study demonstrates that solving ability and authoring ability are partially decoupled. A model can be an excellent solver but a poor problem poser, and vice versa.
Dynamic Difficulty Co-evolution: As stronger models enter the arena, they generate problems that defeat previously dominant solvers. This ensures the benchmark's difficulty scales with participant strength rather than saturating.
Automated Validity Filtering: A robust verification layer that filters out ill-posed questions and corrects erroneous gold answers, ensuring the integrity of the difficulty estimation.
Public Leaderboard: A live leaderboard (mathduels.ai) that updates as new models are released.

4. Experimental Results

The authors evaluated 19 frontier models (from providers like OpenAI, Google, Anthropic, xAI, etc.), generating 570 problems (559 valid after verification) and over 10,000 solve observations.

Ranking Shifts: The best solver is not necessarily the best overall model.
- Example: GPT-5.4-high had the highest Solve Rating (2268), but Gemini-3.1-Pro-high took the top Composite Rank (1919) because it was the strongest Author (Author Rating 1624). Its problems had the lowest average solve rate (62.9%).
Decoupling Evidence: Grok-4.20-high showed the largest gap between solver and author ratings, highlighting that these are distinct axes of mathematical competence.
Difficulty Distribution:
- ~39% of problems were solved by all models (easy), but the remaining "discriminating tail" provided sufficient signal for ranking.
- Problems authored by the two strongest newcomers defeated prior top-3 solvers at a rate of 27.6%, compared to only 8.0% for problems from other participants.
Domain Analysis:
- Discrete Mathematics was the hardest domain (82.5% solve rate) because problems often required discovering hidden structural frameworks rather than applying standard formulas.
- Probability & Statistics was the easiest (92.2% solve rate) as problems often reduced to familiar templates.
Ablation Studies:
- The full 3-step generation pipeline (meta-prompting + amplification) produced significantly harder problems (higher error rates) than direct generation.
- Code execution tools only modestly reduced error rates, confirming the benchmark tests reasoning rather than computation.
- The verifier backbone was robust; different verifier models agreed on problem validity 97.5% of the time.

5. Significance and Future Directions

Beyond Static Benchmarks: MathDuels addresses the fundamental limitation of fixed datasets by creating a "living" benchmark that evolves with the models, ensuring continuous differentiation of capabilities.
Insight into Model Architecture: The decoupling of authoring and solving suggests that current training objectives may not fully align with the ability to generate novel, hard challenges. This has implications for future model training and alignment.
Scalability: The framework is domain-agnostic and can be applied to competitive programming, scientific QA, and other fields where generating hard challenges is a test of understanding.
Future Work: The authors suggest scaling the problem budget to tighten confidence intervals, moving toward proof-based verification to reward rigorous reasoning, and applying the self-play structure to other domains.

In summary, MathDuels represents a paradigm shift from evaluating LLMs as passive solvers to active participants in a competitive ecosystem, providing a more nuanced and sustainable metric for mathematical reasoning capabilities.