Combinatorics · Latin squares · Colourings · Extremal problems
The cells of an \(n \times n\) table are to be coloured with \(n\) different colours in such a way that every row and every column contains cells of all \(n\) colours. Determine the smallest and the largest possible number of pairs of cells of the same colour that have a common vertex,
(a) if \(n = 4\);
(b) if \(n = 5\).
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Serbian Municipal Competition 2021, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source