Move all discs from the first peg to the last
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The Tower of Hanoi is a classic mathematical puzzle invented by French mathematician Edouard Lucas in 1883. It consists of three pegs and a set of graduated discs that can slide onto any peg. The puzzle begins with all discs stacked on the first peg in order of size, largest on the bottom. The goal is to move the entire stack to the last peg, obeying two rules: move only one disc at a time, and never place a larger disc on top of a smaller one.
The elegant recursive solution breaks the problem into three steps: move the top n−1 discs to the spare peg, move the largest disc to the target peg, then move the n−1 discs from the spare peg to the target. This yields a minimum of 2n−1 moves. The legend says monks are moving 64 golden discs between three diamond needles, and when they finish the world will end — at one move per second that would take roughly 585 billion years, far longer than the current age of the universe.
The minimum is 2n−1 where n is the number of discs. So 3 discs need 7 moves, 4 discs need 15, 5 discs need 31, and so on. This formula comes directly from the recursive structure of the solution and has been proven to be optimal.
The simplest approach: on odd-numbered turns, move the smallest disc one peg to the right (wrapping around). On even-numbered turns, make the only legal move that does not involve the smallest disc. This iterative method always produces the optimal solution.
Yes — each disc moves in a predictable cycle. The smallest disc moves every other turn in one consistent direction. Larger discs move less frequently, each appearing at exponentially spaced intervals. This fractal-like pattern makes the solution beautiful and predictable once you see it.