1. the number of ways in the squares of 8x8 chess board can be painted red or blue so that each 2x2 square has two red and two blue squares is

Question

1)2^9

2)2^9-1

3)2^9-2

4)none of these

Answer 1

For an m×n chessboard there are 2^m+2ⁿ−2 ways.

Case I. There are two horizontally adjacent squares of the same color: 2^m−2 ways.

Case II. There are two vertically adjacent squares of the same color: 2ⁿ−2 ways.

Case III. None of the above: 2 ways.

Hint for Case I: There are 2^m−2 ways to color one row so that two adjacent squares have the same color. 

The rest of the coloring is determined from that; colors must alternate in each column. (Note, therefore, that Cases I and II do not overlap.)