The puzzle
Move every ball to the right-hand peg. One at a time, and never a larger ball onto a smaller one. Five rounds, growing from three balls to five.
This is the Tower of Hanoi, invented by Édouard Lucas in 1883 and used ever since as a measure of planning — specifically, the ability to work out a sequence of moves before making the first one.
Scoring rewards the plan, not the persistence
100 × minimum moves ÷ your moves, floored at 30. So finishing is not the
goal — finishing efficiently is. Thrashing your way to a solution in twice the
moves scores half.
The minimum for n balls is 2ⁿ − 1: 7 moves for three balls,
15 for four, 31 for five. That doubling is why the puzzle gets hard so fast, and it is
also the clean recursive structure that makes it solvable by rule rather than by search.
The rule that solves it
The recursive statement is short: to move n balls to the target, move the top n−1 to the spare peg, move the largest to the target, then move the n−1 across. Apply it to itself and the whole solution falls out.
If recursion is not how you think, there is an equivalent mechanical rule that produces the optimal sequence with no planning at all:
- Never move the same ball twice in a row.
- On alternate moves, move the smallest ball — always in the same rotational direction (with an odd number of balls, toward the target peg; with an even number, the other way).
- On the moves in between, make the only legal move that is not the smallest ball. There is always exactly one.
Follow that and you produce the minimum-move solution every time, without ever thinking more than one move ahead. Which is a nice illustration of something general: a good rule can substitute for planning.
Why clinicians and assessments use it
Tower of Hanoi and its cousin Tower of London are standard tasks in executive function assessment, because they require holding a goal, breaking it into subgoals, and inhibiting the move that looks locally helpful but breaks the plan.
The characteristic failure is instructive: people move the largest ball toward the target early because it feels like progress, and then have nowhere to put the smaller ones. That is the same error as optimising the visible step at the cost of the structure.
Frequently asked
Q. I solved it but scored low.
A. Then you took more moves than necessary. Try three balls until you can do it in exactly
seven without thinking, then move up — the pattern is identical at every size.
Q. Is five balls much harder than four?
A. Twice as long (31 moves versus 15), but no harder conceptually. If you have the rule,
size only costs patience.
Q. Is knowing the algorithm cheating?
A. No — recognising that a problem has a general solution is the skill. Nobody solves
five balls by searching.
For a deduction puzzle under a strict budget, try ball weight ordering.