An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 12 black squares, can be drawn on the checkerboard?
It is impossible for a 1x1 square to have 12 black squares
2x2 is also impossible
3x3 is impossible
4x4 is impossible
5x5 will always have at least 12 -> 16 5x5 squares (4*4)
6x6 always have more than 12 -> 9 6x6 squares (3*3)
7x7 always have more than 12 -> 4 7x7 squares (2*2)
8x8 always have more than 12 -> 1 8x8 square (1*1)
Together, we have 16+9+4+1= 30 squares with at least 12 black squares
The answer is 30.
Add to guests answer 4 x 4 IS possible = 25 more
total 30 + 25 = 55 squares
The maximum number of black squares a 4x4 square can have is 8, namely, with 2 in each row, 4 rows. Any other arrangement is impossible. The graph attached supports this.