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Tower of Hanoi
The Tower of Hanoi is a mathematical puzzle invented by French mathematician Édouard Lucas in 1883. It uses three pegs and a set of disks of different sizes stacked in a pyramid. The challenge is to move the whole tower to another peg under strict stacking rules.
Tower of Hanoi Rules
Start with all disks stacked on one peg in order, largest at the bottom and smallest on top. Three pegs are available and the goal is to rebuild the same stack on a different peg.
You may move only one disk per turn, and only the top disk of any peg. A disk can be placed on an empty peg or on top of a larger disk, but never on top of a smaller one.
The puzzle is solved when the entire tower has been transferred to the target peg in the correct size order. The minimum number of moves for n disks is exactly 2ⁿ − 1.
Tower of Hanoi Strategy & Tips
Recurse on the top n−1 disks
To move n disks, first move the top n−1 to the spare peg, move the largest disk to the goal, then move the n−1 stack on top of it. Every solution is this pattern nested inside itself.
Watch the smallest disk's rhythm
The smallest disk moves on every other turn and always travels in the same direction around the three pegs. Keep it cycling and fill the in-between moves with the only other legal move.
There's only ever one good non-small move
On the turns when you don't move the smallest disk, exactly one legal move exists between the other two pegs. Make it — it never requires guessing.
Pick the right starting direction
If the disk count is even, move the smallest disk to the spare peg first; if odd, move it straight to the target peg. Choosing the right first direction guarantees the minimum-move solution.
Frequently Asked Questions
What is the minimum number of moves for the Tower of Hanoi?
For n disks the minimum is 2ⁿ − 1 moves. Three disks take 7 moves, four take 15, and ten disks take 1,023.
How long would 64 disks take?
At one move per second, 2⁶⁴ − 1 moves would take about 585 billion years — the basis of the legend of the priests of Brahma moving a 64-disk tower.
Can you solve Tower of Hanoi with more than three pegs?
Yes. The four-peg version is called the Reve's puzzle and uses the Frame–Stewart algorithm, which needs far fewer moves than the three-peg version.
Is there a pattern to solving it without recursion?
Yes. Always move the smallest disk in one consistent direction every other turn, and make the single other legal move in between. This produces the optimal solution.