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Tower of Hanoi

Move all discs from the left peg to the right peg. You can only place a smaller disc on a larger one.

0
Moves
7
Min Moves
-
Best Moves
Click a peg to select it, then click destination

About Tower of Hanoi Online — Tower of Hanoi Online

Tower of Hanoi online is a classic mathematical puzzle played with three pegs and a stack of discs of graduated sizes. All discs begin stacked on the leftmost peg (Peg A) in size order, largest at the bottom. Your goal is to move the entire stack to the rightmost peg (Peg C), using the middle peg (Peg B) as a temporary resting place. Two rules govern every move: you may only move one disc at a time, and you may never place a larger disc on top of a smaller one. These two simple constraints create a puzzle of elegant mathematical depth.

The Tower of Hanoi puzzle was introduced to the West by French mathematician Édouard Lucas in 1883, who marketed it as a toy and published it under the pseudonym "N. Claus de Siam." Lucas accompanied the puzzle with a legend about a temple in Hanoi where monks were moving a 64-disc tower — and that when they finished, the world would end. The minimum moves required for 64 discs is 2⁶⁴ − 1, roughly 18 quintillion. The puzzle is now a foundational teaching example in computer science for recursive algorithms and binary counting.

Controls

  • Click a peg — Select it and pick up the topmost disc from that peg (the peg rod turns yellow)
  • Click a second peg — Place the held disc onto that peg (only legal moves are accepted)
  • Disc count buttons (3–7) — Set the number of discs and reset the puzzle
  • Reset button — Restart the current disc-count configuration at any time

How to Play Tower of Hanoi Online

The objective of Tower of Hanoi online is to transfer the entire disc stack from Peg A to Peg C in the minimum number of moves.

  • Select a source peg: Click any peg that has discs on it. The selected peg's rod turns yellow to confirm selection, and the top disc is highlighted. If you change your mind, click the same peg again to deselect it.
  • Place the disc: Click a different peg to move the selected disc there. The game only allows legal moves — if the destination peg's top disc is smaller than the disc you are holding, the move is rejected and the status bar explains why. Only moves from smaller-on-top to larger-on-top (or to an empty peg) are accepted.
  • Use Peg B as a buffer: To move a stack of n discs from A to C, you must first move the top n−1 discs to Peg B, move the largest disc directly to Peg C, then move the n−1 discs from Peg B onto Peg C. This recursive structure is the core of every Tower of Hanoi solution.
  • Track the minimum move counter: The "Min Moves" display shows 2ⁿ − 1 for your chosen disc count. For 3 discs this is 7; for 5 discs it is 31; for 7 discs it is 127. Aim to reach this count exactly for an optimal solution.
  • Choose your disc count: Start with 3 discs to learn the recursive pattern quickly. Increase to 4 or 5 once you can solve 3 reliably. Each additional disc exactly doubles the minimum moves required plus one.

The puzzle is complete when all discs rest on Peg C in correct size order with the smallest disc on top.

Tips & Strategies for Tower of Hanoi Online

Apply these techniques to solve Tower of Hanoi online efficiently and understand the underlying structure.

  • Learn the three-disc pattern first: The optimal 3-disc solution is 7 specific moves. Move disc 1 to Peg C, disc 2 to Peg B, disc 1 to Peg B, disc 3 to Peg C, disc 1 to Peg A, disc 2 to Peg C, disc 1 to Peg C. Memorising this 7-move sequence is the foundation — the entire 4-disc, 5-disc, and larger solutions are just this pattern repeated with an additional disc handled around it.
  • Think recursively: Before moving any disc, identify the largest disc that needs to move to Peg C. Everything above it must first be relocated to the buffer peg using the optimal sub-sequence. Then move the large disc, and repeat the sub-sequence to bring the smaller stack on top. This top-down decomposition is the algorithm that produces the minimum-move solution for any disc count.
  • Use the binary counting trick: There is a direct correspondence between Tower of Hanoi moves and binary numbers. For n discs, move number k corresponds to moving the disc whose number equals the number of trailing zeros in the binary representation of k. This mechanical rule generates the complete optimal solution without needing to think recursively and is useful for verifying you are on track.
  • Never move the largest disc unnecessarily: The largest disc moves exactly once in any optimal solution — directly from Peg A to Peg C. If you find yourself contemplating a second move of the largest disc, you have made an error somewhere in the preceding sequence. Backtrack mentally to where the mistake occurred.
  • Track parity for odd/even disc counts: For an odd number of discs, the first move goes to Peg C. For an even number, the first move goes to Peg B. This single rule correctly starts you on the optimal path every time and prevents the first-move uncertainty that beginners often experience.

Skills You Develop Playing Tower of Hanoi Online

Tower of Hanoi online is widely used in computer science education as the canonical example of recursive problem decomposition. Solving it develops the ability to break a large problem into a smaller identical sub-problem, apply the solution recursively, and then reassemble the results — a foundational skill in algorithm design, divide-and-conquer programming, and mathematical induction proofs.

Planning depth is exercised significantly, as every move with n discs requires mentally holding and executing a sequence 2ⁿ − 1 steps long. Working-memory capacity, sequential reasoning, and the ability to track nested subgoals are all exercised directly. These cognitive skills benefit students studying mathematics, logic, or programming, and adults find the puzzle an engaging benchmark for measuring and improving their sequential planning ability.

Frequently Asked Questions about Tower of Hanoi Online

The minimum number of moves to solve n discs is 2ⁿ − 1. For 3 discs it is 7 moves; for 4 discs, 15; for 5 discs, 31; for 6 discs, 63; and for 7 discs, 127. The game displays the minimum for your current disc count and tracks whether you achieve it. An optimal solution is one that matches this exact count exactly — any solution using more moves has at least one redundant or suboptimal move.
Yes — the recursive algorithm solves any disc count. To move n discs from source to destination using a buffer: move the top (n−1) discs from source to buffer; move disc n from source to destination; move the (n−1) discs from buffer to destination. Applying this rule recursively produces the complete minimum-move solution. The base case is moving a single disc directly from source to destination in one move.
No — this is the fundamental constraint of Tower of Hanoi. You may only place a disc on an empty peg or on a disc that is strictly larger. The game enforces this automatically: if you attempt an illegal placement, it rejects the move and displays an error message. This rule is what creates the puzzle's complexity — without it, you could simply move all discs directly to Peg C one by one.
Adding one more disc requires solving the (n−1)-disc problem twice — once to move the smaller stack off the largest disc, and once to move it back on top after relocating the largest disc. Since solving n−1 discs takes 2^(n−1) − 1 moves, and you add one move for the largest disc, the total becomes 2×(2^(n−1) − 1) + 1 = 2ⁿ − 1. This doubling-plus-one recurrence is why each additional disc roughly doubles the difficulty.
The Tower of Hanoi puzzle was introduced by French mathematician Édouard Lucas in 1883. He published it as a toy product and presented it alongside a fictional legend about a temple in Hanoi where monks were working on a 64-disc version. Lucas is known for his work on number theory and primality testing (the Lucas-Lehmer test for Mersenne primes). The puzzle immediately attracted mathematical interest and has been a standard teaching example in recursive algorithm design ever since.
If you click a peg with no discs while selecting a source, the game informs you the peg is empty and no disc is selected. If you click an invalid destination (where the top disc is smaller than the disc you are holding), the move is rejected and the selection is cleared. If you click the same peg you already selected, the selection is cancelled. In all cases the puzzle state is unchanged, so accidental clicks do not count against your move total.
Start with 3 discs. Attempt to solve it by trial and error first — this builds intuition faster than reading the solution. Once you have solved 3 discs a few times, try to reduce your move count to exactly 7. When you can reliably achieve 7 moves, apply the same recursive thinking to 4 discs (minimum 15 moves). Each level reinforces the pattern and prepares you naturally for the next disc count.
Yes — the puzzle is drawn on a responsive canvas that scales to your screen width. Tap any peg column to select or place a disc, exactly as you would click on a desktop. On smaller screens the canvas scales down so all three pegs and the discs remain clearly visible and tappable. The disc count selector and stat counters are also fully touch-accessible above the canvas.