漢諾塔

Tower of Hanoi: rules and solving techniques

Move ordered disks under the classic rules and aim for the minimum moves. This guide reflects the controls and win conditions in the current browser build. It is reviewed by the PocketJoy game-testing team and maintained by Long-Term New Media Limited.

How to play

Three pegs and a stack of discs of different sizes, all starting on the left peg in order from largest at the bottom to smallest at the top. Your task is to move the whole stack to another peg under two rules: only the topmost disc of a peg may be moved, and a larger disc may never rest on a smaller one. Tap the source peg, then the destination peg; illegal moves are simply refused, so the rules police themselves. You can choose how many discs to play with, which adjusts both difficulty and character — each extra disc doubles the theoretical minimum. The game shows that minimum for the chosen size and offers unlimited undo, and the selected peg is visually marked so you never lose track of which one you just picked. On completion your actual move count is shown beside the theoretical minimum.

Strategy

The solution is recursive: to move n discs from A to C, you first move the top n-1 to B using C as temporary storage, move the largest to C, then move the n-1 from B to C using A as storage. Once that structure clicks you need memorise nothing, because every subproblem is identical in form to the original. There is also a more practical rule that produces an optimal solution without thinking recursively at all. If the total number of discs is even, move the smallest disc to the middle peg first; if odd, move it to the destination peg. Then strictly alternate two actions: move the smallest disc, and make the only other legal move available. Note that the second action always has exactly one legal option, since the smallest disc has just moved and cannot move again, leaving only one legal transfer between the other two pegs — so no judgement is required, only execution. The smallest disc's direction follows a pattern too: with an even number of discs it cycles one way, A to B to C to A, and with an odd number it cycles the other. The minimum for n discs is two to the power of n, minus one.

Common mistakes

The usual problem is moving on instinct: the first few steps look like progress and then you start going in circles. The optimal solution is unique, and once you deviate the move count grows quickly — and because every move is legal and nothing warns you, you can wander for dozens of moves without noticing until the counter reveals how far past the minimum you are. The second mistake is starting with too many discs; eight need 255 moves, which is hard to execute without error and harder still to recover from once the rhythm breaks. Begin with three or four, get the alternating rule smooth, then add more. The third is not using undo. Undo is unlimited, so discovering an error means stepping back rather than restarting, yet many players habitually start over and throw away the correct work they had already done. The fourth is forgetting that the roles of destination peg and spare peg keep swapping: at every level of the recursion a different peg serves as storage, and insisting that one particular peg is always the spare is the classic beginner's fixation.

Who it suits

A natural pick for anyone interested in mathematics or algorithms, and one of the classic teaching examples of recursion — nearly every programming course reaches for it, because the recursive structure is clean enough to express in three lines of pseudocode. The puzzle was invented by the French mathematician Édouard Lucas in 1883, who accompanied it with a legend about Brahmin priests moving sixty-four golden discs, with the world ending when they finished; at one move per second, sixty-four discs would take around 585 billion years, so the deadline is not pressing. There is no time pressure, no luck and no hidden information, only logic and execution. Because the disc count is adjustable, it works equally well as a relaxed couple of minutes with three or four discs or a serious attempt at the optimum with seven or eight.

Features

The number of discs is yours to choose, which puts difficulty entirely under your control. The sidebar shows your current move count beside the theoretical minimum for that size, and the gap between them is your room for improvement — this side-by-side display is the single most valuable feature here, because the puzzle has no fail state and without the theoretical figure you would have no way of knowing whether you played well. The completion screen presents both numbers together so you can confirm whether you found an optimal solution. Undo has no limit and steps back a mistaken move without restarting, which is especially useful while learning the alternating rule, since you can try several continuations from the same position. The selected peg is visually marked so you never lose track of which one you picked, which also helps maintain rhythm during long sequences. Restart is always available, and changing the disc count resets automatically. The interface language switches between Traditional Chinese and English with one button.

Controls and local records

Control is a two-step tap: source peg, then destination peg. Illegal moves — a larger disc onto a smaller one — are simply refused, so the rules police themselves and you can concentrate on sequencing without fear of an invalid move. Mouse and touch follow identical logic, the pegs have generous tap areas, and a whole game can be played one-handed on a phone without mis-taps. Undo is unlimited, making mistakes free to reverse. Because there is no timer and no fail state, you can stop and think whenever you like, leave, and come back to continue. Restart is always available. The interface language switches between Traditional Chinese and English with one button, updating every prompt and result string. The game runs locally in your browser, plays offline once loaded, requires no account, and uploads nothing to any server.

Rules FAQ

Reviewed by the PocketJoy game-testing team · Operated by Long-Term New Media Limited · hklongtermad@gmail.com