## Even Ten By Ten

You have a $10\times 10$ checkerboard. How many different squares can be made using an even number of $1\times 1$ squares?
Source: mathcontest.olemiss.edu 1/28/2013

SOLUTION
This problem is similar to the problem titled “Odd Squares” dated 11/6/2006.
A smaller checkerboard $4\times 4$

How many $2\times 2$ squares in a $4\times 4$ checkerboard?
First, we draw a $2\times 2$ square at the upper left corner of the checkerboard. We notice that there are two empty units under it. We move the square down one unit at a time to form two additional $2\times 2$ squares for a total of three. The pattern is $1+\left (4-2\right )=3$.
Likewise, there are two empty units to the right of the original square. We move it one unit at a time to the right to form two additional $2\times 2$ squares for a total of three. The pattern is $1+\left (4-2\right )=3$.
In general, the number of different $x\times x$ squares in an $n\times n$ checkerboard is
$\left [1+\left (n-x\right )\right ]\left [1+\left (n-x\right )\right ]=\left [1+\left (n-x\right )\right ]^2$
A bigger checkerboardĀ $10\times 10$
Let’s use the general formula to calculate the number of different
even squares.
$2\times 2:\left [1+\left (10-2\right )\right ]^2=9^2=81$
$4\times 4:\left [1+\left (10-4\right )\right ]^2=7^2=49$
$6\times 6:\left [1+\left (10-6\right )\right ]^2=5^2=25$
$8\times 8:\left [1+\left (10-8\right )\right ]^2=3^2=9$
$10\times 10:\left [1+\left (10-10\right )\right ]^2=1^2=1$
Total =$81+49+25+9+1=165$

Answer: $165$