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.