Daily Math Puzzle 2026/10/19

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English

Let \(n\) be a positive integer. Start with \(2^n+1\) coins, each in a separate pile. In one move, you may combine two piles, provided that the larger pile contains at most twice as many coins as the smaller one. What is the minimum number of moves combining piles of different sizes needed to gather all the coins into a single pile?

日本語

\(n\) を正の整数とする． \(2^n+1\) 枚のコインを，\(1\) 枚ずつ別々の山にする． 一度に二つの山を合わせられるが，大きい方の枚数は小さい方の \(2\) 倍以下でなければならない． すべてのコインを一つの山にまとめるとき，枚数の異なる二つの山を合わせる操作は，最少で何回必要か．

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