Crowning The Checkers

A well-known problem with checkers is begun by placing eight checkers in a row. A move consists in picking up a checker, carrying it right or left over exactly two checkers, then placing it on a checker to make a king. (Carrying a checker over a king counts as moving it over two checkers.) In four moves form four kings. The problem is not difficult, and it is easy to show that for any even number of checkers, n, when n is at least 8, a row of nil kings can always be produced in n/2 moves.

Numerous variants on this old problem have been proposed by Dudeney and other puzzle inventors. The following variation on the theme, which I believe is new, was suggested and solved by W. Lloyd Milligan of Columbia, S.C.

An even number of checkers, n, are placed in a row. First move a checker over one checker to make a king, then move a checker over two checkers, then a checker over three checkers, and so on, each time increasing by one the number of checkers to be passed over. The objective is to form n/2 kings in n/2 moves.

Can the reader prove that the problem cannot be solved unless n is a multiple of 4, and give a simple algorithm (procedure) for obtaining a solution in all cases where n is a multiple of 4? A solution is easily found by trial and error when n is 4 or 8, but for n= 16 or higher it is not so easy without a systematic method.

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