CayleyPy Beats Classical Algorithms at Solving Large Rubik's Cubes

The CayleyPy project is running a series of competitions on Kaggle focused on finding optimal solutions for mechanical puzzles whose God's number is unknown. Instead of classical algorithms like IDA* and bidirectional BFS, which rely on precomputed Pattern Databases, CayleyPy applies a trainable neural network heuristic. On the classic 3×3×3 Rubik's Cube, with a graph diameter of 20 moves (HTM metric, Tomas Rokicki, 2010), the system achieved 98.4% optimal solutions — surpassing DeepCubeA (60.3%) and EfficientCube (69.6%).

Architecture: three steps from random walks to search

The method consists of three stages. The first is sample generation: starting from a solved cube, the algorithm shuffles it randomly, saving a "configuration–distance from start" pair for each state. The second is training: the neural network learns to predict the distance from a given state, turning it into a search heuristic. The third is Beam Search: the algorithm keeps W of the most promising positions (typically W=5 or W=40), evaluates their neighbors, and selects the top W. The process repeats until the cube is solved or a search limit is reached.

A key finding from the experiments: increasing the training sample size quickly stops improving the heuristic's quality, whereas widening the beam consistently shortens the solution length.

From the 4×4 cube to the

Megaminx: breakthroughs on unknown diameters

On the 4×4×4 cube, the state space contains 7.4×10⁴⁵ states (26 orders of magnitude larger than the 3×3). God's number is unknown; known bounds are 32–53 moves under the BTM metric, with a hypothesis of 48 moves. CayleyPy achieved an average of 46.51 moves on the final leaderboard, beating Tomas Rokicki.

On the Megaminx (dodecahedron), the state space is approximately 1.01×10⁶⁸. The Superflip configuration (all edges flipped, with corners and centers in place) was considered a potential candidate for the configuration farthest from the solved state. Tomas Rokicki solved Superflip in 55 moves under the FTM metric in May 2023. Herbert Kociemba established a lower bound on the Megaminx's diameter in 2016: 48 moves under FTM.

Vladislav Kuznetsov, using CayleyPy, solved Superflip in 47 moves under FTM in May 2026. Since 47<48, it directly follows that Superflip is not the antipode of the Megaminx's Cayley graph, meaning it cannot be a configuration that achieves the diameter. Tomas Rokicki called the result "the first major, significant breakthrough in applying artificial intelligence to the theory of mechanical puzzles" (SpeedSolving.com).

Current competitions: task scale and available resources

Kaggle is hosting competitions for three cubes with unknown God's numbers:

• 5×5×5 cube (2.83×10⁷⁴ states), with a 10,000 ruble prize for solutions shorter than 90 moves; • 6×6×6 cube (1.57×10¹¹⁶ states); • 7×7×7 cube (1.95×10¹⁶⁰ states).

The core problem with classical reinforcement learning here is a sparse signal: the agent only gets useful feedback upon reaching the target state, leaving hundreds of intermediate steps without any feedback. CayleyPy solves this with a continuous signal: the neural network scores every intermediate state with a number, letting the algorithm follow a gradient instead of wandering blindly.

Participants have access to NVIDIA Tesla T4/P100 GPUs and TPU v3-8/v5e-8 on the Kaggle platform, the project's open-source code (480 stars on GitHub), and problems that have gone unsolved for four decades. Anyone can join via Alexander Chervov on Telegram.