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Switching two cards in two boxes

Source: CGMO 2016 Q1

August 13, 2016
combinatorics

Problem Statement

Let n≥3n\ge 3 be an integer. Put n2n^2 cards, each labelled 1,2,…,n21,2,\ldots ,n^2 respectively, in any order into nn empty boxes such that there are exactly nn cards in each box. One can perform the following operation: one first selects 22 boxes, takes out any 22 cards from each of the selected boxes, and then return the cards to the other selected box. Prove that, for any initial order of the n2n^2 cards in the boxes, one can perform the operation finitely many times such that the labelled numbers in each box are consecutive integers.
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