Checkerboard Squares

How many squares that do not contain the shaded cell can be formed on an 8 x 8 checkerboard?

Source: NCTM Mathematics Teacher 2008

SOLUTION
1 x 1 squares
Total number of squares = 8^2; 1 square contains the shaded cell
8^2-1=63 squares do not contain the shaded cell

2 x 2 squares

Total = 7^2; 4 contain the shaded cell in locations 1 through 4
7^2-4=45 squares do not contain the shaded cell

3 x 3 squares

Total = 6^2; 9 contain the shaded cell in locations 1 through 9
6^2-9=27

4 x 4 squares

Total = 5^2; 9 contain the shaded cell in locations 1,2,3,5,6,7,9,10,11
5^2-9=16

5 x 5 squares

Total = 4^2; 9 contain the shaded cell in locations 1,2,3,6,7,8,11,12,13
4^2-9=7

6 x 6 squares

Total = 3^2; all 9 contain the shaded cell in locations 1,2,3,7,8, 9,13,14,15
3^2-9=0

7 x 7 squares

Total = 2^2; all 4 contain the shaded cell in locations 9,10,16,17
2^2-4=0

8 x 8 square
Total = 1^2; 1 contains the shaded cell in location 19
1^2-1=0

Number of squares that do not contain the shaded cell
63+45+27+16+7=158

Answer: 158

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , . Bookmark the permalink.

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