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Tower of Hanoi

Move the stack to peg C one disc at a time, never a large on a small, in the fewest moves: 2^n − 1 from the start, fewer from a scattered one.

  • Logic
  • Untimed
  • 5 levels

How it plays

Discs are numbered by size, 1 the smallest. Each move takes the top disc of one peg to another; a disc may never land on a smaller one. Gather them all on C.

Par is the exact fewest moves. From the classic start that is 2^n − 1; from a scattered start it is found by the same idea — the largest disc out of place moves once, with every smaller disc parked on the third peg. Levels add discs, then scatter the start.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises recursive planning — solving a big case by the same move on a smaller one. It is practice at this puzzle, not a measured gain in planning elsewhere.

Learn

How to get better at tower of hanoi: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Ignore any large discs already sitting in order at the bottom of C. They never need to move.
  2. Find the largest disc that is not yet on C. Every disc smaller than it must first be stacked on the third peg — the one that is neither its peg nor C.
  3. Treat that stacking as the same puzzle, one disc smaller, and solve it the same way.
  4. Move the large disc to C once, then bring the smaller stack onto it as a tower.

Techniques and tricks

  • Move n − 1, then 1, then n − 1

    To move a tower of n discs, move the top n − 1 to the spare peg, the largest to the target, then the n − 1 onto it. Each half is the same puzzle one disc smaller.

    e.g. 3 discs from A to C: the top 2 to B (3 moves), the 3 to C, the 2 onto C (3 moves): 7 in all.

  • The smallest disc goes round

    From the classic start, the smallest disc moves on every other turn, always round the pegs in the same direction. With an odd number of discs it goes A → C → B → A; with an even number, A → B → C → A. Between its moves there is only one legal move that does not touch it.

    e.g. 3 discs: A → C, A → B, C → B, A → C, B → A, B → C, A → C.

  • Double and add one

    From the classic start, the fewest is 2^n − 1. Each extra disc doubles the count and adds one: 7, 15, 31, 63, and 1,023 for 10 discs. Know the par before you begin.

  • Count a scattered start

    Go from the largest disc down, holding a target peg, which starts as C. A disc already on the target adds nothing. A disc that is not adds 2^(k − 1) for disc k, and the target becomes the third peg — neither the disc’s peg nor the old target.

    e.g. Disc 1 on A; discs 3 and 2 on B, 3 underneath; disc 4 on C. Disc 4 is home. Disc 3: + 4, target now A. Disc 2 is on B: + 2, target now C. Disc 1 is on A: + 1. Fewest: 7.

  • Never move the same disc twice running

    Moving a disc and then moving it again is always one move too many: it could have gone straight to where it ended. If you want to, you picked the wrong peg the first time.

As the levels climb

  • You start with three discs from the classic start, all on A. Each level adds a disc, up to the “Most discs” setting, and with “Start” on “Classic, then scattered”, the start scatters after the first disc is added and stays scattered.
  • Each disc doubles the work: five discs is 31 moves, six is 63. Rather than plan the whole line, lean on the smallest-disc rule from the classic start, or the largest-disc-first rule from a scattered one.
  • A scattered start is always at least half as far from done as the classic one. Its par is not 2^n − 1: count it with the scattered rule first, and the plan falls out of the count.
  • With “Extra moves allowed” above 0, par is the fewest plus that many. Solving over par still earns part credit, par divided by the moves you used.

Traps to avoid

  • Sending the smallest disc the wrong way first: with an even number of discs, starting A → C and keeping that cycle builds the tower on B, not C. Check the disc count before the first move.
  • Building the small stack on the wrong peg: the discs above the one you want to move go to the peg that is neither its peg nor the target. Name that peg before you start.
  • Moving a disc that is already home: in a scattered start, a large disc on the bottom of C is in place. Lifting it costs moves you cannot win back.
  • Losing the cycle midway: if you stop to think and forget which way the smallest disc was going, look at the move list above the board: every other move was the smallest disc’s, so the last few show its direction.

How to practise

  • Set “Start” to “Always classic” and “Most discs” to 4 or 5 until the smallest-disc rule runs without thought.
  • Then set “Start” to “Always scattered” and count the fewest before every first move; check your count against the par shown.
  • Use “Extra moves allowed” of 2 or 3 while learning scattered starts, then bring it back to 0.

Levels

It is untimed unless you turn the clock on. There are 5 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More logic

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