Lloyd’s 15

Task number: 2464

Find the criterion for the solvability of the so called Lloyd's 15.

  • Hint

    Consider the following order of the squares, i.e. the position on the left is represented by the permutation: \((1{,}2,3{,}4,8{,}12,14{,}15,13{,}9,5{,}6,7{,}11,10)\).

    Lloyds 15

    Try to derive some rule for such orders.

  • Resolution

    Positions will be represented as numbers of the squares \(1{,}2,…,15\) ordered as they show up on the curve (any other connected curve would be good enough). This is a permutation of \(1,…,15\).

    It suffices to argue that any move preserves the sign of the permutation, i.e. solvable ones have positive sign, while non-solvable negative. Any move shifts a stone between positions with odd and even index (on the curve), i.e. the number of stones that have been passed is always even.

    Note: Samuel Lloyd (1841-1911), found this puzzle in 1878 and announced that \(\$1000\) will be awarded to anyone, who first finds a sequence of moves that interchanges squares 14 and 15. We see that it is not possible, but a tricky solution is at

Difficulty level: Easy task (using definitions and simple reasoning)
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