A problem from the 1996-1997 Estonian Mathematical Olympiad:
A square tabletop measures 3n × 3n. Each unit square is either red or blue. Each red square that doesn’t lie at the edge of the table has exactly five blue squares among its eight neighbors. Each blue square that doesn’t lie at the edge of the table has exactly four red squares among its eight neighbors. How many squares of each color make up the tabletop?
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The tabletop measures 3n × 3n, so we can divide it evenly into n2 3 × 3 squares that together tile the surface completely. For each of these 3 × 3 squares:
- If the unit square at the center is red, there are 5 blue squares and hence 3 red squares adjoining it.
- If the unit square at the center is blue, there are 4 red squares and hence 4 blue squares adjoining it.
That means that each of our 3 × 3 squares is made up of the same constituents: 5 blue squares and 4 red squares. So the 3n × 3n tabletop is made up of 5n2 blue squares and 4n2 red squares.
It remains to show that such a coloring is possible. See the link below, pages 31-32, for a diagram.
(From “The Olympiad Corner,” Crux Mathematicorum 29:1 [February 2003], 22-36.)
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