Tower of Hanoi
Move the stack across. Never put a bigger disc on a smaller one.
The maths Recursion, and why 2ⁿ − 1 grows so fast
No moves yet. The best possible is 7.
Nothing lifted
How to play
Move the whole stack from the left peg to the right peg. Only one disc moves at a time, and a disc may never rest on a smaller one.
Click a peg to lift its top disc, then click another peg to put it down. Clicking the peg you took it from puts it back and costs you nothing. A drop that would break the rule is refused and the status line says why.
Keyboard: left and right arrows move between the pegs, Home and End jump to the outer pegs, Enter or Space lifts and drops, Escape puts a lifted disc back.
Show me one move gives the best move from the position you are actually in. If you have wandered off the shortest route it says so rather than pretending you are still on it.
The starting position never changes, so Reset and New puzzle do the same thing here. Use the Discs control to change the size of the problem: 3 discs need 7 moves, 4 need 15, and 6 need 63. Each extra disc roughly doubles the work.
Why this is mathematics
To move n discs you first move n − 1 out of the way, move the big one, then move the n − 1 back. That self-reference is recursion, and it gives the exact answer: 2ⁿ − 1 moves. Ten discs need 1,023. Sixty-four discs need more moves than there have been seconds since the Big Bang.