Question of the Month Answers: December

December Answer:

Problem: After winning the regional chess tournament, Claire designed a quilt for her bed that had 8 columns across and 8 rows down. With an alternating block pattern, she wanted to make it resemble a chess board. At first glance, Claire's family thought the 64-square quilt was a beautiful addition to her room. Claire knew there were actually many more squares hidden on the quilt. How many squares were there all together?

Solution: At first glance, an 8x8 grid appears to only have 64 squares and one large 8x8 square. While 64 squares is true for the number of 1x1 squares on the grid, there are also many other overlapping squares hidden in the grid.

Below is the number of different squares hidden on the quilt:

	1 x 1 squares	64
	2 x 2 squares	49
	3 x 3 squares	36
	4 x 4 squares	25
	5 x 5 squares	16
	6 x 6 squares	9
	7 x 7 squares	4
	8 x 8 squares	1
	TOTAL	204 total squares on the checkerboard

Interesting patterns: The number of squares is the square of the opposite. For example, there are 64 1x1 squares. 8 x 8, or 8². This pattern continues.