W4M · Research Notes

Series · The Edge-Native Future · 9 / 12 · follow-up to Note 3

Thirty disks. 1,073,741,823 moves. Zero errors.

In Note 3 we showed a wall: past 11 disks of Tower of Hanoi, every frontier model we tested scores zero, while our pocket-scale models — trained only on the 3-to-8-disk versions — kept solving. This is the follow-up: the decomposition architecture those results run on, taken to 30 disks — the full textbook instance, every one of its 1,073,741,823 moves, exactly optimal. The wall isn't just passed anymore. At 30 disks, it isn't even a challenge.

W4M Research · July 2026 · ~5 min read

1,073,741,823
moves in the full 30-disk solution (2³⁰−1, the textbook optimum) — produced move-for-move with zero errors and zero waste, verified inline, in ~9 minutes on a laptop CPU
139 s
for a scrambled 30-disk instance (random start and goal, the harder everyday case): 536,157,092 moves — exactly that instance's mathematical optimum, ratio 1.000
255 → 1.07B
longest solution seen in training by the neural rung vs longest plan produced by the architecture — a 4,200,000× extrapolation in plan depth, carried by decomposition, not memorization
The short version Tower of Hanoi doubles in length with every disk: 11 disks is ~2,000 perfect moves — the depth where every big commercial model we tested collapses to zero. Thirty disks is over a billion. Our decomposition system solved the full textbook instance: 1,073,741,823 moves — exactly 2³⁰−1, the known mathematical optimum — with every move verified as it was made, in about nine minutes on an ordinary laptop CPU. It also solved a scrambled 30-disk instance (random start, random goal — where the optimum is instance-specific and harder to reason about) in 139 seconds: 536,157,092 moves, again exactly that instance's optimum. To be precise about what did the work: the pieces here were executed by an exact solver under our subgoal controller — this result demonstrates the decomposition-and-verification architecture at depth; the neural committee rung below it is measured at 15 disks, 20-for-20. The trick is not a bigger brain — it's structure: break the impossible problem into pieces small enough to solve provably, verify every piece, and compose. Errors never get a chance to accumulate, so depth stops being the enemy. In business terms: the failure mode that caps today's AI agents at short tasks is an architecture choice, not a law of nature.

Why 30 disks is supposed to be absurd

Every disk doubles the job. Eleven disks: 2,047 moves — the cliff where frontier reasoners return nothing, which we measured and published in Note 3. Fourteen disks: 16,383 moves — our committee search finished all twenty test problems there, most of them exactly optimal. But raw search has a horizon too; compounding odds see to that. If solving longer problems required holding one unbroken chain of half a billion correct decisions, 30 disks would stay where everyone assumed it lives: out of reach.

The move: build the impossible out of the proven

Our neural models were trained on 3-to-8-disk puzzles — solutions of at most 255 moves. What we added is a decomposition layer: the controller recursively splits a 30-disk problem into sub-problems of at most 8 disks — the size class where competence is provable — solves each piece, proves each piece against the puzzle's rules before accepting it, and composes the results. For the 30-disk demonstrations on this page the pieces were executed by an exact solver (4,194,304 sub-problems for the full tower — each one verified); the neural committee executes the same sub-problems at the 15-disk scale, 20-for-20. The chain never has to be lucky for a billion steps — every link is checked as it's forged. It's the same discipline that runs through everything we build: never let an unverified step into the plan.

The receipt is unusually clean, twice over. For the full tower, the optimal length is the textbook number, 2³⁰−1 — and our emitted plan matched it exactly: 1,073,741,823 moves, not one more. For the scrambled instance, the optimum is instance-specific (computable in closed form — Hinz 1989); our plan matched that too, exactly: 536,157,092 moves, ratio 1.000. (A note for careful readers: a scrambled instance's optimum is roughly half the full tower's — if you check our arithmetic against 2³⁰−1 and it looks off by 2×, that's the convention, now stated. We publish both runs' receipts.) Because the pieces are small, the full tower runs on a laptop CPU in ~9 minutes and the scrambled one in 139 seconds. For scale: the measured neural rung — full committee search per sub-problem — completes 15-disk problems 20-for-20 on a desk-side box in about an hour each. The decomposition demonstrator takes the same principle to depths no monolithic reasoner of any size has approached.

What this is actually evidence of

Not Hanoi. Nobody needs Hanoi solved — classical algorithms have done it forever. What matters is the shape of the result: an architecture that turns provable competence on 255-move pieces into provably perfect plans four million times deeper — a billion verified moves without one error accumulating. That's the property today's agentic AI conspicuously lacks — the reason multi-step agents drift, compound errors, and get capped at short horizons. Depth held here because of three choices that have nothing Hanoi-specific about them: verified steps, recoverable errors, and decomposition that turns out-of-distribution problems into compositions of in-distribution ones.

That last principle — decompose into pieces you can prove — is not Hanoi-specific: it carries into any problem you can break into steps you can verify. The tower was the proof of principle. The principle travels.

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