You have a checkerboard. How many different squares can be made using an even number of squares?
Source: mathcontest.olemiss.edu 1/28/2013
This problem is similar to the problem titled “Odd Squares” dated 11/6/2006.
A smaller checkerboard
How many squares in a checkerboard?
First, we draw a 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 squares for a total of three. The pattern is .
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 squares for a total of three. The pattern is .
In general, the number of different squares in an checkerboard is
A bigger checkerboard
Let’s use the general formula to calculate the number of different even squares.