Long-Jumping Checkers

Number Theory Level pending

A game is played on a standard \(8\times 8\) checkerboard. Checkers are placed on one half of the checkerboard. Label the squares like an array, with squares \((i,j)\) for \(1\le i,j\le 8\) Checkers are then moved with the following rule:

Choose a checker at square \((i,j)\). Pick another square \((m,n)\); then, the checker can move to the spot \((3m-2i,3n-2j)\) if it is vacant.

Let the number of possible checker arrangements be \(N\times 10^M\), where \(10\not\mid N\) and \(M\) is a non-negative integer. Find the last three digits of \(N+M\).

Details and Assumptions

The starting position counts as one arrangement. Move one checker, and it's another arrangement. Your job is to find the total number of arrangements with any number of moves.

Rotations and reflections count as different arrangements.

You may use a calculator to calculate the value of \(N\times 10^M\).

The starting position has checkers in squares \((i,j)\) for \(1\le i \le 8\) and \(1\le j\le 4\).


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