Where's the maths? This puzzle is a machine for discovering recursion: solving a problem by first solving a smaller copy of itself. Writing that down gives a recurrence relation — a rule for the number of moves in terms of the number for one fewer disc — and solving the recurrence gives a formula with a power of 2 in it. Hidden inside one wooden toy are exponential growth, binary numbers, and proof by induction: the way you show a formula is right not just for 5 discs but for every possible number of discs.

To move n discs from one peg to another you must first move the top n−1 discs out of the way, shift the big disc, then move those n−1 discs back on top. So the puzzle solves itself — it just needs a smaller copy of itself twice.

That gives the recurrence T(n) = 2·T(n−1) + 1, with T(1) = 1, and unrolling it gives the closed form T(n) = 2ⁿ − 1.

• 5 discs → 31 moves. 8 discs → 255 moves. Each extra disc doubles the work.
• The legend: 64 golden discs in a temple, one move a second. 2⁶⁴ − 1 seconds is about 585 billion years — the universe is only ~13.8 billion years old.
• Number the moves 1, 2, 3, … In the shortest solution, move k always shifts the disc given by the number of trailing zeros in k written in binary, plus one. Move 4 = 100₂ has two trailing zeros, so it moves disc 3. Watch the counter.
• Draw every legal position of a 3-disc puzzle as a dot and join positions one move apart: you get the Sierpiński triangle. The shortest solution is one side of it.