Combinatorics · Tilings · Colouring arguments · Invariants · Powers of two
A \(2004 \times 2004\) board is completely tiled by pieces of size \(1 \times 4\); each piece covers four cells of a single row (call it horizontal) or four cells of a single column (vertical). Can the number of horizontal pieces be equal to the number of vertical pieces?
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Serbian Republic Competition (Republicko takmicenje) 2004, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source