Tower of Hanoi

You start with a neat stack of disks on one peg, largest at the bottom. Your goal is to rebuild that stack on another peg. The rules are strict: move one disk at a time, and never put a larger disk on top of a smaller one. That single constraint is what makes the puzzle hard.

To solve it well, you cannot just react to the board in front of you. You have to look several moves ahead and set up positions that pay off later. That is why this task measures planning and executive function: your ability to hold a goal in mind, break it into steps, and resist the tempting move that leads to a dead end.

• Goal management: keeping the final stack in mind while you work on sub-goals.
• Look-ahead: anticipating where a disk needs to be two or three moves from now.
• Inhibition: avoiding the quick move that blocks your own path.

The Tower of Hanoi is a mathematical puzzle with a tidy structure. For a stack of n disks, the shortest possible solution is exactly 2 to the power of n, minus one moves. So three disks need 7 moves, four disks need 15, and five disks need 31. Every extra disk roughly doubles the work, which is why the puzzle scales so sharply.

In cognitive research, this puzzle is a close relative of the Tower of London task, a planning test built on the same idea of moving objects between pegs under rules. Both are used to study how people plan ahead and manage sub-goals. The clean math of the Tower of Hanoi is useful because there is a known optimal move count, so your solution can be compared directly against the perfect one.

Two numbers tell the story here: how many moves you used, and how long you took. The gold standard is the minimum move count for the number of disks. On four disks, that target is 15 moves. Beating that is impossible; the interesting question is how close you get and how quickly.

• Efficient: at or very near the minimum, with steady, deliberate moves.
• Average: a few extra moves as you correct course partway through.
• Struggling: many extra moves, usually from solving move by move instead of planning.

Speed and move count trade off. Racing tends to add wasted moves, while long pauses to plan can lower your move count but raise your time. Neither is "correct"; they just reflect different styles. Compare your own runs rather than a universal target.

This puzzle rewards a strategy, and once you learn it your scores can improve a lot. The key insight is recursion: to move a stack of four, you first move the top three out of the way, move the biggest disk, then move the three back on top. Seeing that pattern turns a confusing puzzle into a repeatable routine.

• Work from the bottom up: think about where the largest disk must go, then clear a path for it.
• Use the spare peg on purpose: the middle peg is temporary storage, not a mistake.
• Plan the first few moves before touching anything: a short pause up front saves many wasted moves.

Be honest about the gains, though. Much of your improvement is a practice effect and pattern learning, not a broad jump in planning ability. There is also a hard ceiling: you cannot beat the minimum move count, so once you hit it, only your time can improve. And with regression to the mean, one especially clean run is often followed by a more ordinary one.

The most common error is solving reactively: grabbing whatever disk you can legally move without thinking about the consequences. That feels like progress but often blocks the disk you actually needed to move next.

• Deflates your score: moving the same disk back and forth, or shuffling without a plan.
• Deflates your score: ignoring the spare peg and cornering yourself.
• Inflates your score unfairly: memorizing the exact move sequence from a previous run and replaying it from memory.
• Adds wasted time: long hesitation on every single move instead of planning in short chunks.

Keep the disk count and device consistent between attempts. Then a lower move count and a faster time reflect a genuinely better plan, not a memorized answer.