The Token Games: Evaluating Language Model Reasoning with Puzzle Duels

We introduce Pencil Puzzle Bench, a framework for evaluating large language model reasoning through pencil puzzles, a family of constraint-satisfaction problems closely related to NP-complete problems, with deterministic, step-level verification. From a database of 62,231 puzzles across 94 varieties with verified unique solutions, we select a benchmark of 300 puzzles spanning 20 varieties and evaluate 51 models from 11 providers in two modes: direct ask (single-shot) and agentic (multi-turn with iterative verification). A key differentiator of our benchmark is that every intermediate board state can be checked against variety-specific constraints, localizing errors to the exact rule violated, providing the infrastructure for dense, per-move reward signals for process supervision and reinforcement learning. Our evaluation reveals two distinct axes of capability: (1) reasoning effort scaling, where GPT-5.2 improves 81x from no reasoning to maximum effort; and (2) agentic iteration, where Claude Opus 4.6 rises from 0.3% to 30.0% through iterative checking, while GPT-5.2@xhigh improves from 20.2% to 56.0%. Agentic attempts span a median of 29 turns over 17 minutes, with the longest exceeding 1,221 turns and 14.3 hours - a demanding test of long-context utilization, not just reasoning.

As frontier language models attain near-ceiling performance on static mathematical benchmarks, existing evaluations are increasingly unable to differentiate model capabilities, largely because they cast models solely as solvers of fixed problem sets. We introduce MathDuels, a self-play benchmark in which models occupy dual roles: each authors math problems under adversarial prompting and solves problems authored by every other participant. Problems are produced through a three-stage generation pipeline (meta-prompting, problem generation, and difficulty amplification), and validated by an independent verifier that excludes ill-posed questions. A Rasch model (Rasch, 1993) jointly estimates solver abilities and problem difficulties; author quality is derived from the difficulties of each model's authored problems. Experiments across 19 frontier models reveal that authoring and solving capabilities are partially decoupled, and that dual-role evaluation reveals capability separations invisible in single-role benchmarks. As newer models enter the arena, they produce problems that defeat previously dominant solvers, so the benchmark's difficulty co-evolves with participant strength rather than saturating at a fixed ceiling. We host a public leaderboard that updates as new models are released.

Large language models often appear strong on symbolic and algorithmic tasks, yet this apparent strength can hide brittle behaviour when problems become longer, harder, or slightly out of distribution. A major limitation of current reasoning benchmarks is that many primarily test whether a model can produce a valid answer, while paying less attention to whether the solution is minimal, robust, and stable under controlled difficulty scaling. We introduce RecurrReason, a difficulty-controlled benchmark of four recurrent logic puzzles (Tower of Hanoi, River Crossing, Block World, and Checkers Jumping) with BFS-optimal trajectories and a single interpretable difficulty parameter $N \in \{1,\dots,10\}$, totalling 10{,}817 unique puzzles and 285{,}933 moves. We benchmark two Transformer families, an encoder-decoder model (T5-style) and a decoder-only model (GPT-2-style), under consistent data splits and evaluation criteria, training on $N{=}1$ to $7$ and evaluating on both held-out in-distribution instances and harder out-of-distribution instances at $N{=}8$ to $10$. Fine-tuned pre-trained T5 achieves 97.27% validation and 81.00% OOD accuracy on Block World; all models score 0.00% on River Crossing under all conditions. Failure mode analysis reveals that architecture is a stronger determinant of success than scale. Pre-training transfers only to puzzles with locally structured transition functions. Our code and dataset will be open-sourced upon acceptance.