Sort ten alternating hats using only paired moves
Ten hats hang on a row of twelve pegs — silk, felt, silk, felt, alternating — with two empty pegs at one end. Moving two adjacent hats at a time onto the two empty pegs, without ever reversing their left-right order, crossing your hands, or moving just one hat, end up with all five silk hats together and all five felt hats together, empty pegs again at one end.
Reveal the answer
It takes exactly 5 paired moves. Numbering the pegs 1–12 left to right, one solution moves the hats at positions 2-3, then 7-8, then 4-5, then 10-11, then 1-2. Henry Dudeney published this as 'The Hat Puzzle,' no. 236 in Amusements in Mathematics (1917), as a warm-up for a tougher ten-counter puzzle he set right after it.
— Henry Ernest Dudeney, Amusements in Mathematics — 1917, puzzle no. 236
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