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There is a 64-square grid (8 x 8 squares). There is a piece that can start in any square. It can only move 3 spaces in a straight line and then move 1 space to the right or the left (the type of L-movement a knight could make on a Chess board but one space farther than a knight before turning). Can this piece end up in a square that is adjacent to its original square?

I think not, but I am not 100% sure. And if it can't, I am having trouble proving this mathematically. Is it just that if it only can move 3 spaces in a straight line, and the board is 8 squares long, you will eventually end up diagonally from the original square?

Thanks!

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The answer is

NO

Because

In one step it always goes to the same color of its originating position. As in checkerboard the adjacent cells have different color then it's impossible.

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