## 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: $158$