Coloring A Big Board
Consider a board in which its squares have been colored red or blue. We know that for each blue square, that is not on the edge, 4 of the 8 squares that are adjacent, are red. Also, we know that for each red square, that is not on the edge, 5 of the 8 squares that are adjacent, are blue. Find the maximum number of red squares on the board.
Details and Assumptions :
2 squares are adjacent if they share at least a vertex. Indeed, each square that's not on the edge has exactly 8 adjacent squares.
The answer is 360.
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1 solution
We know that in each 3 × 3 square there will be 4 red squares and 5 blue squares, because if in the center is a red square, the edges will always have 5 blue squares, and if in the center is a blue square, the edges will have 4 red squares:
R B R B R R B B B
R B R B B R B R B
As a 2 7 × 3 0 board can be divided in ninety 3 × 3 squares, the number of red squares in the board is 9 0 × 4 = 3 6 0 .
Certainly, a board colored with these conditions can be achieved simply by doing this:
Take, for example, this 3 × 3 arrangement:
R B R B R R B B B
3 0 times to form the 2 7 × 3 0 board.
Join 2 of these matrices side by side successively, and we have colored all the board. An interesting question would be how many different boards are with these conditions?