Tower of Hanoi

Move the tower from the first peg to the third, one disc at a time, never a bigger disc on a smaller one. 3 to 8 discs, and an owl who knows the fewest moves.

Tower of Hanoi rules

  1. Move the whole tower from peg 1 to peg 3.
  2. Move one disc at a time: the top disc of any peg.
  3. Never put a bigger disc on a smaller one.

How to solve it

The fewest moves

A tower of n discs takes at least 2n − 1 moves: 7 for 3 discs, 15 for 4, 31 for 5, 63 for 6, 127 for 7 and 255 for 8.

The plan

Move every disc but the biggest to peg 2, the biggest to peg 3, then the rest onto it. Each smaller tower is the same puzzle with one disc fewer.

Disc 1's rhythm

Disc 1 moves every other turn, always the same way round: peg 1, 3, 2, 1 with an odd number of discs, peg 1, 2, 3, 1 with an even number. Between disc 1's moves, there is only one move you can make that isn't disc 1: make it.

All 15 moves for 4 discsThe rules in full