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n x n square and strawberries

Source: IMO Shortlist 2006, Combinatorics 4, AIMO 2007, TST 4, P2

June 28, 2007
matrixcombinatoricsPartial OrdersIMO Shortlist

Problem Statement

A cake has the form of an n n x n n square composed of n2 n^{2} unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement A\mathcal{A}.
Let B\mathcal{B} be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement B\mathcal{B} than of arrangement A\mathcal{A}. Prove that arrangement B\mathcal{B} can be obtained from A \mathcal{A} by performing a number of switches, defined as follows:
A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
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