Combinatorics · Chessboard · Colouring invariant · Covering
A bishop on a chessboard attacks every square lying on one of the two diagonals through it. Call a square covered if a bishop stands on it or a bishop attacks it.
Prove that seven bishops can never be placed on a standard \(8 \times 8\) chessboard so that every square of the board is covered, but that eight bishops can.
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Serbian Municipal Competition 2012, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source