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

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About mvtrinh

Retired high school math teacher.
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