Tower of Hanoi with 10 Disks

The shape of the solution

There are never 1,023 separate moves to remember — only three stages:

1. Moves 1 to 511: build a tower of 9 disks on B.
2. Move 512: disk 10 crosses to C — the only time it moves.
3. Moves 513 to 1,023: rebuild the 9-disk tower on top of it.

Stages 1 and 3 are each the 9-disk solution, with two towers' names swapped.

Which disk moves when

Disk 1 makes half of all the moves — 512 of them, on every odd-numbered move. Each larger disk moves half as often as the one above it, which is the binary counter hiding inside the puzzle: the disk that moves is one more than the number of times you can halve the move number and still get a whole number. Move 6 halves once, to 3, so move 6 is disk 2. There is more on that in Tower of Hanoi and binary.

The smallest disk's circuit

With 10 disks — an even number — disk 1 travels A → B → C → A for the whole game, never reversing. Every move in between is the only legal move that leaves disk 1 alone. Those two rules alone reproduce the list above move for move; the iterative solution explains why, and strategies turns it into something to play by.

About the 10-disk puzzle

1,023 moves is the number a 1966 Doctor Who serial built its climax on. In The Celestial Toymaker the Doctor is made to play the Trilogic Game, a ten-piece Tower of Hanoi that has to be completed in exactly 1,023 moves. Whoever wrote the rules had done the arithmetic: 2¹⁰ − 1 is the minimum for ten pieces, so the game demanded a perfect solution and nothing less.

Played by hand at a move a second, ten disks is seventeen minutes of unbroken attention, and the difficulty is no longer working out what to do — the rule for the next move is the same as it was at three disks — but not losing your place. The checkpoints that help are the finished towers: nine disks on B after move 511, and disk 10 alone crossing on move 512.

Ten is the largest size with a page like this one. Past it, the solver lists every move up to twelve disks and plays solutions up to fifteen, and the calculator takes the count as far as you like. In binary, 1,023 is ten ones; it factors as 3 × 11 × 31. Ten is even, so disk 1 goes to B first.

Other sizes

• 9 disks511 moves — the solution that makes up each half of this one.
• 64 disksThe size of the legend: 18,446,744,073,709,551,615 moves.
• Solve 10 disks from a position you are stuck inThe solver finishes a half-played game in the fewest moves left, rather than starting again.
• Every solution, from 3 to 10 disksMove counts, first moves and timings side by side.

Frequently asked questions

How many moves does it take to solve Tower of Hanoi with 10 disks?

1,023 moves is the minimum: 2¹⁰ − 1 = 1,023. No solution with fewer moves exists, and any solution with more has wasted some.

What is the first move in 10-disk Tower of Hanoi?

Move disk 1, the smallest, from tower A to tower B. With an even number of disks the smallest disk travels A → B → C → A for the whole game, so its first stop is B. Sending it to C first still lets you finish, but it costs one extra move.

How long does 10-disk Tower of Hanoi take to solve?

A perfect game is 1,023 moves, which at one move a second is 17 minutes and at three moves a second — about as fast as anyone plays by hand — 5 minutes.

Which Doctor Who story features the Tower of Hanoi?

The Celestial Toymaker, first broadcast in 1966. The Doctor has to win the Trilogic Game, a ten-piece Tower of Hanoi that must be completed in exactly 1,023 moves — which is 2¹⁰ − 1, the minimum for ten disks.