# Liquid Reasoning Transformers: A Sudoku-Based Prototype for Chess-Scale Algorithmic Tasks

arXiv: https://arxiv.org/abs/2512.12792 · HTML: https://mlsanity.com/paper/2512.12792/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 12 references that are in the map.
- First posted on arXiv 14 Dec 2025.

## Builds on (most cited first, 7 of 7)

- [Evidence of Learned Look-Ahead in a Chess-Playing Neural Network](https://mlsanity.com/paper/2406.00877/) (2024)
- [Thinking Tokens for Language Modeling](https://mlsanity.com/paper/2405.08644/) (2024)
- [Complete Chess Games Enable LLM Become A Chess Master](https://mlsanity.com/paper/2501.17186/) (2025)
- [Implicit Search via Discrete Diffusion: A Study on Chess](https://mlsanity.com/paper/2502.19805/) (2025)
- [Chess AI: Competing Paradigms for Machine Intelligence](https://mlsanity.com/paper/2109.11602/) (2021)
- [Deep Autoregressive Models as Causal Inference Engines](https://mlsanity.com/paper/2409.18581/) (2024)
- [The Algorithm Is Not the Behavior: Learned Priors Override Look-Ahead in a Chess-Playing Neural Network](https://mlsanity.com/paper/2508.21380/) (2025)

## Abstract (arXiv)

The Liquid Reasoning Transformer (LRT) is a transformer architecture designed for inference with adaptive depths using iterative changes, discard-based correction, and a learned stopping mechanism. Instead of relying on a single feedforward pass, the model updates a recurrent reasoning token across multiple internal steps, allowing it to correct early errors and allocate computation based on input difficulty. We evaluate the LRT on Sudoku as a controlled testbed for structured reasoning and show that it achieves strong performance, reaching 98.68% digit accuracy and 36.30% full-puzzle accuracy without using symbolic rules or search. Analyzing internal patterns shows that the discard and stop gates play different, important roles in stabilizing inferences and adjusting computational depth. We discuss how these mechanisms extend naturally to chess-scale reasoning tasks and outline extensions for multi-token reasoning and larger domains.