Towers of Hanoi · AdAstraMath

The Puzzle

Three pegs — A, B, C — and a stack of discs on peg A, largest at the bottom. Move every disc to peg C.

The Rules

1

Only one disc can be moved at a time.

2

Only the top disc of a peg can be picked up.

3

A disc may only land on an empty peg or on a larger disc.

Minimum Moves

With n discs the minimum number of moves is always:

2ⁿ − 1

Discs (n)	Minimum moves
3	7
4	15
5	31
6	63

The Strategy (Recursion)

To move n discs from A to C using B as a spare:

1

Move the top n−1 discs from A → B (using C as spare).

2

Move the largest disc from A → C.

3

Move the n−1 discs from B → C (using A as spare).

Each step is the same puzzle with one fewer disc — a classic example of recursion. The move count satisfies T(n) = 2·T(n−1) + 1, giving T(n) = 2ⁿ − 1.

Example: 3 Discs (7 moves)

#	From → To	Disc
1	A → C	Small
2	A → B	Medium
3	C → B	Small
4	A → C	Large
5	B → A	Small
6	B → C	Medium
7	A → C	Small

Keyboard Shortcuts

Press a key to select/move the top disc of that peg:

1 — Peg A 2 — Peg B 3 — Peg C R — Reset