Tetris Block Puzzle is a single player stochastic puzzle in which a player places blocks on an 8 x 8 grid to complete lines; its popular variants have amassed tens of millions of downloads. Despite this reach, there is little principled assessment of which rule sets are more difficult. Inspired by prior work that uses AlphaZero as a strong evaluator for chess variants, we study difficulty in this domain using Stochastic Gumbel AlphaZero (SGAZ), a budget-aware planning agent for stochastic environments. We evaluate rule changes including holding block h, preview holding block p, and additional Tetris block variants using metrics such as training reward and convergence iterations. Empirically, increasing h and p reduces difficulty (higher reward and faster convergence), while adding more Tetris block variants increases difficulty, with the T-pentomino producing the largest slowdown. Through analysis, SGAZ delivers strong play under small simulation budgets, enabling efficient, reproducible comparisons across rule sets and providing a reference for future design in stochastic puzzle games.
Figures & tables
Figure 1: The gameplay of Tetris Block Puzzle. (a) The player chooses an O-tetromino from three holding blocks and places it (highlighted with a red border), completing a vertical line. This clears the line and earns the player 1 point. After the O-tetromino is taken, the first preview holding block, an S-tetromino, becomes a new holding block for the next turn. Then, a new preview holding block is randomly selected from the standard blocks shown in (b) and may be rotated.
Figure 2: The training curve for the classic game with h=3 and p=0 .
Figure 3: The training curve for p=0 with different h .
h
1
2
3
Reward
39.0
4126.1
6544.0
Table 1: The training rewards of the different numbers of holding blocks.
h
1
2
3
Convergence speed
-
160
61
Table 2: Convergence iterations for different numbers of holding blocks. The “-” mark indicates that the agent did not converge within training iterations.
Figure 4: The training rewards impact of different numbers of preview holding blocks. The blank areas are expected to be optimal, as their preceding settings are already optimal.
Figure 5: The training curve of different numbers of preview holding blocks under a fixed setting of holding blocks h=1 .
Figure 6: The training curve of different numbers of preview holding blocks under a fixed setting of holding blocks h=2 .
Figure 7: Impact of different numbers of preview holding blocks on convergence iterations. The mark of “X” area indicates that the agent did not converge within training iterations.
Figure 8: The U,V,X,T-pentomino blocks.
Figure 9: The training rewards impact of adding additional blocks.
Figure 10: The convergence iterations impact of adding additional blocks.
Recursive reasoning models such as the Tiny Recursive Model (TRM) iteratively refine two latent states: an answer state y, decoded into the output, and a reasoning state z, which is never decoded. Many of their failures have been attributed to bad basins, regions of latent space that trap the recursion on wrong answers. Recent test-time approaches perturb y, z or both to recover these failures, but the impact of this choice on performance, test-time compute efficiency and scaling properties remains unclear. Through controlled experiments on TRM, we find that a failed trajectory carries its failure in its answer state. At equal compute, in depth and in width, resampling y recovers more failures than any perturbation of z we tested, and the performance gap widens as the test-time compute budget grows. Our approach, Gated Answer Resampling (GAR), pairs this resampling of the answer state with the model's own halting head and restarts only the puzzles that the head rejects. With at most 64 restarts per puzzle and no training or hyperparameter search, GAR takes TRM from 85.9% to 99.4% on Sudoku-Extreme and from 68.8% to 85.6% on Pencil Puzzle Bench validation. On the same checkpoints, GAR also outperforms Probabilistic TRM (PTRM), which perturbs z at each step, even when PTRM uses 42 to 197 times as much compute.
Maxime Illouz
Concordia University · Mila – Québec AI Institute · Montréal, Canada
Tile-matching puzzle games serve hundreds of millions of players, yet the content-generation algorithms that decide which pieces to present at each turn remain proprietary, and no open platform exists for studying adaptive difficulty in this genre. We present an adaptive tile-matching platform whose central algorithmic contribution is a dual-track content-generation architecture: a deterministic rule-based generator that is always available, and an optional learned generator, both subject to a common verification gate that establishes, by exhaustive sequential-placement search, that every delivered piece set is fully placeable so the learned track can never degrade the constructive-feasibility guarantee of the rule track. A self-play reinforcement-learning placement agent, supervised by auxiliary tasks that expose per-shape placeability to shared representations, is used to diagnose the game's dominant failure mode: at high board fill, long-bar pieces lose the majority of their legal placements. Across 234,000+ self-play episodes the agent reaches a 35.6% win rate (median score 4,200), and controlled simulation shows that at board fill rates of 70--75%, 33--56% of long-bar pieces have no legal placement, while spawn difficulty distributions are statistically indistinguishable between won and lost games---evidence that board-state degeneration, not content difficulty, drives late-game failure. Head-to-head ablations show that per-shape placeability supervision---not aggregate difficulty features---drives the representation gain, and a 14-day online gray rollout (48,000 players; sample-ratio verified, CUPED-adjusted) lifts day-1 retention by 1.8 percentage points and session duration by 7% over the rule track alone, quantifying the neural track's asymmetric upside in live play.
We introduce PuzzleJAX, a GPU-accelerated puzzle game engine and description language designed to support rapid benchmarking of tree search, reinforcement learning, and LLM reasoning abilities. Unlike existing GPU-accelerated learning environments that provide hard-coded implementations of fixed sets of games, PuzzleJAX allows dynamic compilation of any game expressible in its domain-specific language (DSL). This DSL follows PuzzleScript, which is a popular and accessible online game engine for designing puzzle games. In this paper, we validate in PuzzleJAX several hundred of the thousands of games designed in PuzzleScript by both professional designers and casual creators since its release in 2013, thereby demonstrating PuzzleJAX's coverage of an expansive, expressive, and human-relevant space of tasks. By analyzing the performance of search, learning, and language models on these games, we show that PuzzleJAX can naturally express tasks that are both simple and intuitive to understand, yet often deeply challenging to master, requiring a combination of control, planning, and high-level insight.
Sam Earle, Graham Todd, Yuchen Li +5
New York University · University of Malta · University of the Witwatersrand +1