CRAM Solves Sudoku in 42 Milliseconds and Beats Looped Reasoning Models
A learned energy function inspired by Hopfield and Tank cracks Sudoku-Extreme in 42 milliseconds, exposing weak reasoning benchmarks along the way.
- CRAM, a trainable Hopfield-Tank energy, solves Sudoku-Extreme in 42 ms training time on one GPU.
- Matches looped reasoning models at up to 1,000x less training cost and 282x fewer inference FLOPs.
- Trained without backpropagation through time by pushing energy down on solutions, up elsewhere.
- Reaches ~100% Sudoku-Extreme accuracy with just 30 puzzles vs 700+ for looped baselines.
- Hand-coded energies with 3 parameters solve all four benchmarks, questioning their value as reasoning tests.
- Needs an extra hidden layer to handle Maze-Hard; not yet a general reasoning model.
A new preprint introduces CRAM, or Compact Reasoning Associative Memory, a trainable formulation of the Hopfield and Tank energy model first published in 1985. The authors report that CRAM matches or exceeds looped reasoning models on Sudoku and maze benchmarks while training at least 1,000 times faster and using at most 1/282 as many inference FLOPs at matched accuracy.
The result challenges what strong performance on these benchmarks demonstrates. Sudoku-Extreme and Maze-Unique can apparently be solved by learning compact constraints over valid answers, which limits their value as evidence for architectures designed to perform long, iterative reasoning.
Why loops cost so much
Looped reasoning models repeatedly apply the same network to a hidden state, refining that state until it represents a solution. This approach has performed well on structured puzzles, but each additional iteration adds computation and creates another step through which training signals must pass.
Backpropagation through time typically trains these models by differentiating through every unrolled solver step. The resulting workload consumes memory, increases training time, and can make optimization sensitive to the number of iterations. It also asks the model to learn a sequence of state transitions from puzzle to answer.
Reasoning as an energy surface
CRAM assigns a scalar energy to each complete candidate grid. Training lowers the energy of valid solutions relative to invalid assignments, creating a surface whose low points correspond to configurations that resemble the training data. Pairwise weights capture which cell values tend to coexist in valid solutions.
Inference begins by fixing the supplied clues, randomly initializing unknown cells, and updating one cell at a time to reduce the total energy. Several chains run in parallel from different random states. Noisy updates help them leave local minima, and the solver stops when any chain crosses an energy threshold fitted from training data.
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