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Toppling Tower of Hanoi

Master the Tower of Hanoi with mathematical insights and optimal solving strategies.
By Puzzuzu TeamAug 16, 2025
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The Tower of Hanoi looks deceptively simple—just move a stack of disks from one peg to another. But this classic puzzle, invented by French mathematician Édouard Lucas in 1883, conceals one of the most elegant mathematical patterns in all of puzzle-dom. While anyone can understand the rules in seconds, the mathematical implications can be quite fascinating. The bottom line: there's exactly one optimal solution path, no matter how many disks you're juggling and you can easily find it once your understand what you're looking for.

The Deceptive Simplicity

The rules couldn't be simpler: move all disks from the starting peg to the destination peg, but never place a larger disk on top of a smaller one. With just three disks, this feels trivial. With seven disks? You're looking at 127 moves if you do it optimally—and thousands more if you don't.

Lucas originally called his creation the "Tower of Brahma" and crafted a legend around it: monks in a temple had 64 golden disks that they needed to transfer following these exact rules. When they finished, the world would end. Thankfully, even working around the clock, it would take them over 580 billion years to complete the task.

The Mathematical Foundation

Here's where the Tower of Hanoi reveals its mathematical soul. For n disks, the minimum number of moves required is always 2^n - 1. This isn't a rough estimate—it's mathematically proven to be exact.

  • 3 disks: 7 moves
  • 4 disks: 15 moves
  • 5 disks: 31 moves
  • 6 disks: 63 moves
  • 10 disks: 1,023 moves

This exponential growth explains why the puzzle becomes dramatically more challenging with each additional disk. What feels manageable with 4 disks becomes a serious time investment with 7 or 8. Each time you add one disk, you're essentially doubling the previous solution and adding one move. The solution for n+1 disks involves solving the n-disk problem twice, plus one move for the largest disk.

The Recursive Pattern

The beauty of the Tower of Hanoi lies in its recursive nature. To move n disks from peg A to peg C:

  1. Move the top n-1 disks from A to B (using C as temporary storage)
  2. Move the largest disk from A to C
  3. Move the n-1 disks from B to C (using A as temporary storage)

This pattern holds regardless of the number of disks. The same three-step process works whether you're moving 3 disks or 30. Each step follows the identical logic, just applied to progressively smaller sub-problems.

Optimal Solving Strategy

There are a few simple ways you can look at solving a basic game of Tower of Hanoi. This simple iterative approach is probably the easiest for beginners to grasp.

First, determine the rotation direction: The direction depends on the number of disks:

  • Even number of disks: Move the smallest disk clockwise around the pegs
  • Odd number of disks: Move the smallest disk counter-clockwise around the pegs

Once you've established the direction:

  • Starting at the first turn, the smallest disk moves every other turn in this predetermined direction
  • On alternate turns, make the only legal move not involving the smallest disk

Step by Step approach to solving the puzzle

  • As you can see in the image above the smallest disk moves first to the third tower. That's because there are an odd number of disks and so it moves counter-clockwise.
  • Next, only the middle-sized disk can move to the second tower - That is the only legal move.
  • Now, the smallest disk moves to second stack because it is time to move the smallest disk again and it is moving counter-clockwise.
  • The only legal move is to move the largest piece counter-clockwise to the third tower.
  • The pattern continues moving odds then the biggest piece until the puzzle completes.

Advanced Variations

With four pegs or more, the problem becomes significantly more complex—and the optimal solution wasn't verified until 2014. The Frame-Stewart algorithm can be applied to any Tower of Hanoi puzzle with more than three pegs, such as Reve's puzzle. Here's how it works:

For p pegs and n disks, choose a split size k with 1 ≤ k < n:

  • Move the top k disks to a spare area using all p pegs.
  • Move the remaining n−k disks to the target using only p−1 pegs (one peg is occupied by the k stack).
  • Move the k disks from the spare area onto the target using all p pegs again.

If T(p, n) is the minimum number of moves with p pegs and n disks, then for any k the plan above takes 2·T(p, k) + T(p−1, n−k) moves. The Frame–Stewart algorithm picks the k that makes this quantity as small as possible. For p = 4 (Reve’s puzzle) this strategy was proved optimal in 2014; for p > 4 it’s the best known method and widely believed to be optimal, but not fully proved.

If this doesn't make sense well then try reading On the Frame–Stewart algorithm for the multi-peg Tower of Hanoi problem by Sandi Klavžar, Uroš Milutinović, and Ciril Petr. Then it will make less sense.

Ready to Test Your Skills?

Understanding the mathematics is one thing, but developing the intuitive feel for optimal moves requires practice. Start with smaller configurations and work your way up, paying attention to the patterns your movements create.

Try the Tower of Hanoi puzzle and see if you can spot the mathematical patterns in action. Once you've internalized the recursive structure, you'll find yourself solving configurations that once seemed impossibly complex.

Remember: every optimal solution follows the same elegant mathematical rules, regardless of size. You're not just moving disks—you're executing one of the most beautiful algorithms in all of mathematics.

Additional Resources

For those interested in diving deeper into the Tower of Hanoi: