Moving 3 disks is something you can succeed at after just a few tries. But once you get to 5 or 7 disks, it suddenly takes much longer, and by 10 disks, the number of moves needed is almost too many to count by hand. Why does adding just one more disk make things so much harder, so suddenly? The answer is that the Tower of Hanoi is a problem that grows "exponentially."
The rules are simple. There are three pegs, and disks stacked in size order on the first peg need to all be moved to the third peg. There are exactly two conditions: you can move only one disk at a time, and you can never place a larger disk on top of a smaller one. For n disks, the minimum number of moves needed is exactly 2ⁿ − 1. That's 1 move for 1 disk, 3 for 2 disks, 7 for 3 disks, 15 for 4 disks… the required number of moves nearly doubles every time you add one more disk.
Why exactly 2ⁿ−1? To move n disks, you first move the top n−1 disks (everything except the big bottom one) to the spare peg — which alone already takes 2ⁿ⁻¹−1 moves — then move the bottom disk to the target peg (1 move), and finally move the n−1 disks from the spare peg back onto the target peg (another 2ⁿ⁻¹−1 moves). Written as an equation, that's (2ⁿ⁻¹−1) + 1 + (2ⁿ⁻¹−1) = 2ⁿ−1. This approach — "breaking a big problem down into a smaller problem of the exact same shape" — is called recursion. The Tower of Hanoi is the most famous example for learning to think recursively, and it's a staple example when computer science students first learn to program.
There's a famous legend about an Indian temple tied to this puzzle. Monks are said to be moving 64 golden disks by this very rule, and the world will end once they finish. The minimum number of moves for 64 disks is 2⁶⁴−1 — even moving one disk per second without stopping, that would take about 585 billion years, far longer than the age of the universe itself (about 13.8 billion years). Thanks to this legend, the Tower of Hanoi is often cited as a classic example of just how terrifyingly fast exponentially growing quantities can explode.
This kind of exponential growth matters well beyond the Tower of Hanoi too. Bacteria doubling in number at regular intervals, and a sheet of paper becoming impossibly thick after just a few folds — both work on the same principle. Few examples build an intuitive feel for "a small-looking rule, repeated, growing far beyond what you'd imagine" as effectively as the Tower of Hanoi.
When trying this with kids, start with 3 disks and have them solve it in the minimum number of moves (7), then move up to 4 disks (15) and 5 disks (31), keeping track of "how many moves it took" each time. Seeing for themselves that the required number of moves grows to a little more than double with each extra disk makes the abstract concept of exponential growth land far more vividly.
On our activity page, you can choose a difficulty from 3 up to 6 disks and try it yourself, comparing your move count against the theoretical minimum in real time. Try challenging yourself to solve it in exactly the minimum number of moves.