MathDB

Let

n>1

be an integer. A rook stands in one of the cells of an infinite chessboard that is initially all white. Each move of the rook is exactly

n{}

cells in a single direction, either vertically or horizontally, and causes the

n{}

cells passed over by the rook to be painted black. After several such moves, without visiting any cell twice, the rook returns to its starting cell, with the resulting black cells forming a closed path. Prove that the number of white cells inside the black path gives a remainder of

1{}

when divided by

n{}

Problem Statement