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array of squares

Source: 30-th Vietnamese Mathematical Olympiad 1992

February 17, 2007
geometryrectanglecombinatorics proposedcombinatorics

Problem Statement

Label the squares of a 1991×19921991 \times 1992 rectangle (m,n)(m, n) with 1≤m≤19911 \leq m \leq 1991 and 1≤n≤19921 \leq n \leq 1992. We wish to color all the squares red. The first move is to color red the squares (m,n),(m+1,n+1),(m+2,n+1)(m, n), (m+1, n+1), (m+2, n+1)for some m<1990,n<1992m < 1990, n < 1992. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?
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