Lloyd’s 15

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: $$(1{,}2,3{,}4,8{,}12,14{,}15,13{,}9,5{,}6,7{,}11,10)$$.

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
http://www.holotronix.com/samlloyd15c.html.