Among the myriad challenges used to evaluate artificial intelligence, few are as deceptively simple as the Towers of Hanoi. With its three pegs and a stack of disks, it appears to be a straightforward exercise in recursive logic, a puzzle that a computer science undergraduate solves in an afternoon. Yet, in the context of AI research, this classic problem is not merely a test of algorithmic implementation; it is a profound probe into the architecture of reasoning itself. It exposes the fundamental differences between statistical pattern matching, reinforcement learning, and explicit planning, revealing the chasm between simulating intelligence and actually possessing it.
When we observe an LLM solve a Towers of Hanoi puzzle, the surface-level success can be misleading. The model outputs a sequence of moves that appears correct, often explaining the recursive strategy: move the top \(n-1\) disks to the auxiliary peg, move the largest disk to the target, and then move the \(n-1\) disks onto the largest. However, the mechanism behind this generation is fundamentally different from the algorithmic process a computer executes. The LLM is not running a recursive function; it is predicting the next token based on a probability distribution derived from vast amounts of training data that likely include descriptions of the puzzle and its solutions. The “reasoning” is an illusion of pattern recognition, a high-dimensional statistical correlation rather than a causal chain of logic.
The Illusion of Understanding in Large Language Models
Large Language Models (LLMs) operate by compressing the internet into a set of weights and biases. When presented with a Towers of Hanoi prompt, the model accesses latent representations of the puzzle’s structure. If the puzzle is small—say, three disks—the probability of generating the correct sequence is high because the solution space is small and the pattern is common in the training corpus. The model generates text that mimics the output of a recursive algorithm, but it lacks an internal state or a world model that tracks the physical constraints of the disks.
This distinction becomes critical as the complexity scales. For a three-disk configuration, the optimal solution requires \(7\) moves. For four disks, \(15\) moves. For \(n\) disks, \(2^n – 1\) moves. As the number of disks increases, the length of the output sequence grows exponentially. LLMs have a limited context window and a tendency to hallucinate or lose coherence over long sequences. Without the ability to maintain a precise, symbolic representation of the current state of every disk, the LLM eventually fails. It might repeat moves, violate the rule that a larger disk cannot sit on a smaller one, or simply stop generating valid steps.
What does this failure reveal? It demonstrates that LLMs are not reasoning engines in the classical sense. They are sophisticated autocomplete systems. When the problem requires deep, multi-step planning that falls outside the distribution of their training data, or when the solution length exceeds their effective memory, the statistical approximation breaks down. The failure is not a lack of knowledge; the model “knows” the rules. The failure is a lack of grounding—the ability to simulate the consequences of actions in a consistent, internal model.
Reinforcement Learning and the Struggle for Generalization
Reinforcement Learning (RL) approaches the Towers of Hanoi from a completely different angle. Instead of being fed the solution, an RL agent explores the environment through trial and error, guided by a reward signal. In a standard setup, the agent receives a positive reward for moving a disk to the target peg and a negative reward for illegal moves or excessive steps. The agent learns a policy—a mapping from states to actions—that maximizes cumulative reward.
Early RL agents, particularly those using tabular methods or simple linear function approximators, struggle with the combinatorial explosion of the state space. For \(n\) disks, there are \(3^n\) possible configurations. As \(n\) grows, the number of states becomes unmanageable for exhaustive exploration. The agent must generalize. This is where the comparison between RL and LLMs becomes fascinating.
RL agents trained on a small number of disks (e.g., 3 or 4) often learn a specific sequence of moves that works for those specific configurations. When presented with a larger number of disks (e.g., 6 or 8), they frequently fail to generalize the recursive strategy. They have memorized a trajectory, not learned the underlying invariant: that the problem can be decomposed into smaller sub-problems. This is a classic example of interpolation versus extrapolation. RL agents excel at interpolating within the distribution of states they have visited, but they struggle to extrapolate to novel, larger configurations.
However, modern RL techniques, particularly those utilizing deep neural networks and techniques like Hindsight Experience Replay (HER), have shown promise. By reframing failures as successful demonstrations of what not to do, or by artificially setting the goal state to whatever was achieved, agents can learn more robust policies. Yet, even state-of-the-art RL agents often require millions of episodes to solve the 4-disk Hanoi, whereas a symbolic algorithm solves it instantly. This highlights a key difference: RL is a process of discovery, while symbolic planning is a process of derivation.
Symbolic Planners: The Precision of Logic
Symbolic AI, or “Good Old-Fashioned AI” (GOFAI), treats the Towers of Hanoi as a planning problem defined by logic. In this paradigm, the problem is modeled using a formal language like PDDL (Planning Domain Definition Language). The state is defined by predicates such as on(disk, peg) and clear(disk). The actions are defined by preconditions (e.g., you can only move a disk if it is clear and the destination is clear) and effects (e.g., moving disk A updates the state to remove on(A, peg1) and add on(A, peg2)).
When a symbolic planner is tasked with solving Towers of Hanoi, it does not guess. It uses search algorithms—such as A* (A-star) or Dijkstra’s algorithm—to traverse the state space. The planner builds a graph of possible states and finds the shortest path from the initial state (all disks on peg A) to the goal state (all disks on peg C). Because the Towers of Hanoi has a mathematically proven optimal solution length of \(2^n – 1\), symbolic planners are guaranteed to find this optimal path, provided they have enough memory and time.
The performance of symbolic planners on Hanoi is deterministic and perfect. They do not hallucinate, and they do not need to “learn” the strategy. The strategy is encoded in the domain definition. This reveals a fundamental truth about reasoning: explicit representation of rules and states allows for guaranteed correctness. However, the scalability is limited by computational resources. As \(n\) increases, the branching factor of the search tree (roughly 1.5 for Hanoi) leads to an exponential increase in memory requirements. Yet, for the range of \(n\) that humans typically care about (up to 20 or so), symbolic planners are vastly superior to statistical models.
Comparing these three approaches—LLMs, RL, and Symbolic Planners—paints a clear picture of the trade-offs in AI architecture. LLMs offer flexibility and natural language interface but lack strict logical consistency. RL offers adaptability to complex environments but requires immense data and struggles with generalization. Symbolic planners offer precision and optimality but require explicit modeling and suffer from scalability issues in highly complex, unstructured domains.
What Failures Reveal About Reasoning Depth
When an AI fails at the Towers of Hanoi, the nature of the failure is diagnostic. It tells us exactly where the system’s reasoning breaks down.
If an LLM fails by producing an illegal move (e.g., placing a larger disk on a smaller one), it indicates that the model’s attention mechanism failed to attend to the specific constraints of the current state. The model is generating text based on local context and general patterns rather than maintaining a global consistency check. It is “thinking” word by word, not state by state.
If an RL agent fails by getting stuck in a loop—moving a disk back and forth between two pegs—it reveals a lack of exploration or a poorly shaped reward function. The agent has learned a local optimum but cannot escape it. This is a failure of credit assignment: the agent cannot trace the long-term consequence of a current action to a distant future reward. In Hanoi, the reward for moving the smallest disk is immediately positive, but the reward for solving the puzzle is far away. The agent must learn to delay gratification, a concept that RL struggles with without sophisticated techniques like reward shaping or discount factors.
If a symbolic planner fails due to memory exhaustion, it reveals the limitations of brute-force search. While the planner “understands” the problem perfectly, it lacks the heuristic insight to prune the search tree effectively beyond the basic rules. This highlights the difference between search and heuristic reasoning. A human expert solving Hanoi doesn’t search the entire tree; they apply a recursive heuristic that collapses the search space. Symbolic planners can incorporate heuristics, but defining them for arbitrary problems remains a challenge.
These failures underscore that “reasoning” is not a monolithic capability. It comprises several distinct components: state tracking, constraint satisfaction, long-term planning, and generalization. The Towers of Hanoi acts as a crucible that separates these components. It forces an AI to reveal whether it is merely manipulating symbols (LLM), reacting to stimuli (RL), or logically deducing a path (Symbolic Planner).
The Turing Test and the Chinese Room
The Towers of Hanoi also serves as a modern instantiation of Searle’s Chinese Room argument. Imagine an LLM inside a room. You slide a question about Hanoi under the door. The LLM, following a complex set of rules (its weights) that it does not understand, manipulates the symbols and slides back a correct sequence of moves. To the observer outside, the LLM appears to understand Chinese—or in this case, the logic of Hanoi. But does it?
The answer depends on your definition of understanding. If understanding is defined by the ability to produce correct outputs, then yes. But if understanding requires an internal model of the world that mirrors the causal structure of reality, then no. The LLM has no concept of a “disk” or a “peg.” It has vector embeddings that represent these words, but it cannot simulate the physics or the logic of the puzzle without generating text.
RL agents come closer to a embodied understanding. They interact with the environment (even if simulated) and receive feedback. They learn that moving a disk has consequences. However, this is still a shallow form of understanding, often lacking the abstract generalization capabilities of a human child who, after seeing a 3-disk puzzle, can instantly reason about a 10-disk puzzle.
Symbolic systems possess a formal understanding. They know the rules explicitly. However, they lack the flexibility to handle ambiguity or noise. If the input is slightly perturbed—for example, if a disk is described as “medium-sized” rather than strictly defined by its diameter—the symbolic system may fail to initialize the state correctly.
Benchmarks of the Future: Beyond Hanoi
While the Towers of Hanoi is a classic, the AI community is constantly seeking benchmarks that test reasoning in more complex, real-world scenarios. Hanoi is fully observable, deterministic, and discrete. The real world is often partially observable, stochastic, and continuous.
However, the principles tested by Hanoi remain relevant. We are seeing a convergence of architectures. Neuro-symbolic AI combines the pattern recognition of neural networks with the logical reasoning of symbolic systems. In such a system, a neural network might perceive the state of the puzzle (recognizing the disks and pegs from an image), while a symbolic planner determines the next move. This hybrid approach attempts to capture the best of both worlds: the flexibility of learning and the precision of logic.
Furthermore, the evaluation of Large Language Models is shifting from simple accuracy to “reasoning depth.” Benchmarks now include multi-step reasoning problems that require the model to maintain a chain of thought. The Towers of Hanoi, when framed as a reasoning challenge, tests the model’s ability to decompose a problem. Can the LLM generate a recursive algorithm? Can it execute that algorithm symbolically? Recent research suggests that LLMs can indeed simulate recursion to some extent, but they are prone to errors as the depth increases.
The study of these systems through the lens of Hanoi teaches us patience. It reminds us that intelligence is not a single metric but a spectrum of capabilities. We often marvel at the fluency of LLMs, but Hanoi reminds us of their brittleness. We admire the endurance of RL agents, but Hanoi highlights their inefficiency. We respect the accuracy of symbolic planners, but Hanoi exposes their rigidity.
As we push the boundaries of AI, the lessons from this simple puzzle persist. True intelligence likely requires a synthesis of these approaches: the statistical intuition of LLMs, the adaptive exploration of RL, and the rigorous logic of symbolic reasoning. Until we build systems that can seamlessly integrate these modes of thought, the Towers of Hanoi will remain a humble but unforgiving judge, silently waiting to see if our creations can truly think, or merely compute.
The journey through these three paradigms—LLMs, RL, and Symbolic Planning—reveals that the Towers of Hanoi is more than a child’s game. It is a mirror reflecting the current state of artificial intelligence. It shows us where we have succeeded in mimicking the outputs of intelligence and where we still lack the fundamental processes of understanding. For the engineer and the scientist, it remains an essential tool for dissecting the anatomy of thought, one move at a time.
