5th Centromerican 2003 Problem B2 An 8 x 8 board is divided into unit squares. Each unit square is painted red or blue. Find the number of ways of doing this so that each 2 x 2 square (of four unit squares) has two red squares and two blue squares.
2 9 - 2. Solution Thanks to José Nelson Ramirez We can choose the colors in the first column arbitrarily. If the squares in the first column alternate in color, then there are 2 choices for the second column, either matching the first column, or opposite colors. Similarly, there are 2 choices for each of the remaining columns. There are 2 ways in which the squares in the first column can alternate in color, so we get 2 8 ways in all with alternating colors in each column. If the squares in the first column do not alternate in color, then there must be two adjacent squares the same color. Hence the two squares adjacent to them in the second column are determined. Hence all the squares in the second column are determined. It also has two adjacent squares of the same color, so all the squares in the third column are determined, and so on. There are 2 8 ways of coloring the first column. 2 of these ways have alternating colors, so 2 8 - 2 have two adjacent squares the same. 5th OMCC 2003 © John Scholes jscholes@kalva.demon.co.uk 2 Dec 2003 Last corrected/updated 2 Dec 03