Overview
This research investigates how Transformers, from small, in-house models to large, third-party Large Reasoning Models (LRMs), address the Tower of Hanoi puzzle, specifically focusing on the flat-to-flat variant. The study explores the mechanistic failures observed in these models, particularly concerning the maintenance of emergent internal representations.
Research Context
The Tower of Hanoi is a planning puzzle previously identified as challenging for LRMs. While current models can solve the standard formulation, the flat-to-flat variant, which does not restrict initial and goal states to a single peg, continues to pose difficulties. Prior work has reported a performance collapse in LRMs on such tasks.
Approach
The study employed a two-pronged approach. First, small, in-house Transformers were trained from scratch using precomputed solution traces for the Tower of Hanoi puzzle. A variety of interpretability techniques were then applied to analyze the internal workings of these models. Second, the researchers extended their analysis to frontier reasoning models, specifically Qwen3.6-27B and DeepSeek-R1-Distill-Qwen-32B. These large models' performance on the flat-to-flat Tower of Hanoi variant was evaluated, particularly when attempting to solve the task through extended chain-of-thought. To understand the failures, representations were probed at different stages during planning, and causality was established by attempting to improve performance through prompt-time representation injection at inference.
Findings
- Small Transformers trained on Tower of Hanoi solution traces develop an emergent world model.
- This emergent world model is a linearly decodable, geometrically faithful representation of the puzzle's state space, corresponding to the Sierpinski triangle.
- The emergent world model is causally involved in the small Transformers' puzzle-solving process.
- Large LRMs, including Qwen3.6-27B and DeepSeek-R1-Distill-Qwen-32B, encode the Sierpinski world model near-perfectly by the end of the prompt.
- Despite encoding the world model, these large LRMs fail at the majority of tasks when the puzzle involves more than 3 rings.
- The failure in large LRMs is attributed to the decaying representation of the world model during the planning process.
- Probing revealed a degradation of the world model representation at different stages of planning.
- Injecting the prompt-time representation into the models during inference can improve performance.
- The observed performance collapse in large LRMs is not due to an absence of a world model but rather a failure to maintain the required representations over time.
- Performance is at least partially recoverable through intervention.
Why This Matters
This research reframes the understanding of performance limitations in Large Reasoning Models on complex planning tasks. It suggests that such models are capable of forming internal world models but struggle with the sustained maintenance of these representations, indicating a target area for future model improvement. The ability to partially recover performance by re-injecting representations highlights a potential pathway for enhancing the reliability and reasoning capabilities of these advanced AI systems.