BMO Round 2, 2025

Combinatorics Difficulty 6.0 National Olympiad Find the answer United Kingdom

An $n \times n$ chessboard consists of $n^2$ cells which are unit squares. Each cell is coloured black or white so that cells with a common edge are different colours. Isaac muddles up the colouring by repeatedly swapping either two complete columns or two complete rows. Elijah wants to restore the original colouring by repeatedly swapping either two complete columns or two complete rows.

In terms of $n$ , what is the largest number of swaps that Elijah might need?

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