Combinatorics · Tilings · Boards · Double counting
For natural numbers \(m\) and \(n\), consider a board of dimensions \(m \times n\) made up of \(mn\) unit squares. Call the skeleton of the board the set of all unit segments that are edges of at least one of these unit squares.
Determine all ordered pairs \((m, n)\) of natural numbers for which the skeleton of the \(m \times n\) board can be tiled by copies (rotations allowed) of the figure consisting of two perpendicular unit segments with a common endpoint - that is, the skeleton can be split into such pieces so that every unit segment of the skeleton belongs to exactly one piece.
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Serbian Municipal Competition 2023, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source