Tower of Hanoi Variations: 4-Peg, Cyclic, Bicolor and More

The classic Tower of Hanoi has three pegs and one simple rule about disk sizes, but mathematicians have spent over a century inventing twists on it. These Tower of Hanoi variations change the number of pegs, the directions disks can move, or what counts as a legal stack, and each one produces a fresh puzzle with its own optimal strategy. This guide tours the most interesting variants, from the famous 4-peg Reve's puzzle to cyclic and bicolor versions. If you're new to the original, start with how to solve the Tower of Hanoi.

The 4-peg puzzle (Reve's puzzle)

The best-known variation simply adds a fourth peg. With an extra spare peg, you can solve the puzzle in fewer moves than the classic 2ⁿ − 1, because there's more room to park disks. This version is often called Reve's puzzle.

Finding the optimal solution turned out to be surprisingly deep. The leading method is the Frame-Stewart algorithm, proposed independently in the 1940s, which works by moving some top group of disks aside using all four pegs, then moving the rest with the classic three-peg method, then bringing the first group back. For decades it was only conjectured to be optimal; the proof that Frame-Stewart gives the true minimum for four pegs wasn't completed until 2014. Adding pegs beyond four generalizes the same idea, with the math getting harder at each step.

Cyclic Tower of Hanoi

In the cyclic variation, the three pegs are arranged in a circle and disks may only move in one direction around it, say always clockwise. You can't move a disk straight from peg A to peg C if that means going "backwards." This restriction makes the puzzle longer and changes the strategy, since reaching the peg you want often means going the long way around the cycle. The minimum move count grows compared to the standard puzzle because of the lost flexibility.

Bicolor and "Tower of Hanoy" variants

The bicolor version doubles up the disks: instead of n distinct sizes, you have pairs of disks in two colors, and the goal often involves separating or alternating the colors. The rules about not stacking larger on smaller still apply within sizes, but the color goal adds a sorting challenge on top of the moving challenge.

Related puzzles swap disks for other objects entirely. Some "Tower of Hanoy" card-game variants use playing cards in place of disks, keeping the one-at-a-time, ordered-stack spirit while changing the components. These tend to be novelty takes rather than serious mathematical objects, but they show how flexible the core idea is.

Magnetic and constrained variants

The magnetic Tower of Hanoi gives each disk two faces (like magnet poles) and adds a rule that two like faces can't touch, so a disk sometimes flips as it moves. This small change forces extra moves and creates a genuinely different solving rhythm.

There are also variants that restrict which pegs connect to which, effectively turning the three pegs into a path or other graph instead of a fully connected triangle. The linear (or "three-in-a-row") Tower of Hanoi only allows moves between adjacent pegs, so you can never jump from the first peg to the third directly. Like the cyclic version, this lengthens every solution.

Why the variations matter

Beyond being fun, the variations are a small window into how mathematicians work. Take a solved problem, change one rule, and ask: does the old strategy still work? What's the new minimum? The 4-peg puzzle is the standout example, a tweak so natural it seems trivial, yet its optimal solution resisted proof for nearly 75 years. That's a good reminder that simple-looking puzzles can hide serious depth, which is part of the Tower of Hanoi's enduring appeal. The puzzle's place in math and computing is covered more in what the Tower of Hanoi teaches.

Start with the classic

Most of these variations build directly on the standard three-peg solution, so the best preparation is to master the original first. Once the recursive pattern is second nature, the twists become much easier to reason about. Play the classic across all five sizes, from the 3-disk warm-up to the 10-disk Einstein challenge, and you'll have the foundation every variation is built on.