Is the Tower of Hanoi an IQ Test? What It Measures in People and in AI

The Tower of Hanoi has three pegs and a stack of disks that must be moved, one at a time, from the first peg to the third without ever placing a larger disk on a smaller one. People search for "tower of hanoi iq test" and "tower of hanoi 4 disks iq", and since June 2025 they also search for "apple ai tower of hanoi". The short answers are that the puzzle has no IQ scale, that psychology uses it as a measure of planning and self-control, and that in AI research it became the centrepiece of a dispute about whether "reasoning" models really reason. We read the original Apple paper, the published reply and Apple's revised version.

How many moves does a Tower of Hanoi take?

The French mathematician Édouard Lucas first presented the puzzle in 1883. The minimum number of moves for n disks is 2^n – 1, and that is why the puzzle is both easy to state and hard to finish at scale. Doubling the disks does not double the work, it squares it.

• 3 disks: 7 moves; 4 disks: 15 moves; 5 disks: 31 moves
• 8 disks: 255 moves; 10 disks: 1,023 moves
• 15 disks: 32,767 moves; 20 disks: 1,048,575 moves
• Lucas's legend of 64 disks: 18,446,744,073,709,551,615 moves, or about 585 billion years at one move per second

Once the rule is known, every step is mechanical. A well-known iterative method moves the smallest disk one peg in a fixed direction on odd-numbered moves and makes the only other legal move on even-numbered ones. That is the first reason the puzzle is a poor IQ item at large sizes: with four disks it tests planning, and with ten it tests whether you can carry out 1,023 steps without a slip.

Is the Tower of Hanoi an IQ test?

No, and no norm sample exists for it. An IQ is a position among people on a test with a stated mean, standard deviation and reliability, which is how IQ tests work; Hanoi times and move counts have none of that. What the research literature does show is overlap with other abilities. In an individual-differences study of 137 college students (Miyake and colleagues, Cognitive Psychology, 2000), Tower of Hanoi performance was related most strongly to the inhibition of prepotent responses, whereas the Wisconsin Card Sorting Test was related to shifting. In a study of 85 undergraduates (Zook and colleagues, Brain and Cognition, 2004), working memory, inhibition and fluid intelligence each accounted for significant variance on the Tower of Hanoi, and working memory and inhibition kept doing so after fluid intelligence was taken into account.

So the puzzle is partly a fluid-reasoning task and partly something else. Neither abstract gives an effect size we can turn into points, so any table that does is invented. Our explainer on executive function versus IQ covers why planning and inhibition tests correlate with, but are not, intelligence tests, and working memory and reasoning are not the same covers the memory side.

What did Apple's Tower of Hanoi study find?

Parshin Shojaee, Iman Mirzadeh, Keivan Alizadeh, Maxwell Horton, Samy Bengio and Mehrdad Farajtabar submitted "The Illusion of Thinking" to arXiv on 7 June 2025 (version 3 is dated 20 November 2025). They gave models four controllable puzzles: Tower of Hanoi, Checker Jumping, River Crossing and Blocks World, raising the size step by step. The reasoning models with visible thinking traces were o3-mini (medium and high settings), DeepSeek-R1, DeepSeek-R1-Distill-Qwen-32B and Claude 3.7 Sonnet with thinking. Each puzzle size was run 25 times per model, with up to 64,000 tokens allowed for Claude 3.7 Sonnet and for DeepSeek on local servers, and an answer counted only if every move was right.

The headline result was an accuracy collapse. Standard models beat reasoning models on easy sizes, reasoning models won in the middle, and both fell to near zero beyond a threshold; near the threshold the reasoning models spent fewer thinking tokens even though budget remained. For Tower of Hanoi the paper places the collapse at 7 and 8 disks, which is 127 and 255 moves. In a test where the model was given the solution algorithm in the prompt, so that it only had to execute it, performance did not improve and the collapse came at roughly the same point. The paper also reports runs of up to about 100 correct moves in Hanoi but only 4 in River Crossing with three pairs, an 11-move puzzle, and notes that Claude 3.7 Sonnet with thinking was near-perfect on 5 disks (31 moves) yet failed the three-pair River Crossing. The authors link the gap to how familiar each puzzle is from training data.

What did the published reply say?

A. Lawsen's "The Illusion of the Illusion of Thinking" (arXiv, 10 June 2025, version 2 dated 16 June) made three arguments. First, a model that lists every move of a large Hanoi runs into output limits. Using an estimate of 10 tokens per move, 12 disks need roughly 41,000 tokens and 13 disks roughly 82,000, so with a 64,000-token ceiling the reply puts the largest feasible size at 12 or 13. Second, River Crossing with six or more pairs and a boat that holds three people has no solution, yet those cases were scored as failures. Third, when models were asked for a short program that prints the solution, rather than the 32,767 moves, they did well at 15 disks, finishing in under 5,000 tokens.

Two points in the reply matter more than they first appear. The author's own token arithmetic puts the output ceiling at 12 or 13 disks, which is well above the 7 to 8 where Apple reports collapse, and the reply notes this. And the impossible-instance point is a real error in the River Crossing setup, not a quibble.

What changed in Apple's revised paper?

The November 2025 version contains a question-and-answer section that responds to critics. On River Crossing it accepts that the puzzle changes character at six or more pairs and narrows its claim to fewer than six, arguing that collapse already appears at three pairs. On Tower of Hanoi it says it removed sizes above 12 to address context-limit concerns, and it argues that the collapse at 7 to 8 disks sits well inside the context window because the first wrong move typically comes long before the end: for 8 disks, around move 40 to 50 of 255. It also reports a temperature-zero run on DeepSeek-R1 to show that sampling is not the main cause. Our reading of the two papers is that they agree on the arithmetic and disagree on interpretation: whether failing to write out 255 moves reliably is a reasoning limit, an execution limit, or both.

What does this say about "AI IQ"?

It says that puzzle benchmarks are sensitive to format. A model asked for the full move list and a model asked for a program that generates it are being tested on different skills, and a headline built on one format can reverse on the other. That is the same lesson as in our reports on why AI benchmark scores disagree and on a PNAS reasoning benchmark. And it is why a rescaled puzzle count is not an IQ, as we explain in when a model "scores 120", that number is not an IQ. For a harder, less familiar test see ARC-AGI-3, and for a count of right answers turned into an "IQ" see AI scores on the Mensa Norway test.

• What format did the answer take: a full list of 2^n – 1 moves, or a method?
• Were all test cases solvable? The River Crossing error shows why this must be checked
• Was the algorithm given, and did execution still fail? Apple says yes at 7 to 8 disks
• How many runs per cell: Apple used 25, the reply's Lua test too few to power a test
• Is the number a rank among people, or a count of right answers rescaled?

Our IQ Metrics test reports a percentile among people, and the IQ percentile calculator shows what any score means as a rank. For the wider question of machines and human intelligence, see is AI smarter than humans, and for whether models share a general factor, read is there a g factor for AI models.

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