February 2007
Problem of the Month

How Many Squares?
by Bob Stanton
Games Magazine February 2007

In this 4x4 grid of circles, how many different sets of four circles form the corners of a square (of any size)? One such set is shaded for you.

Solution to the Problem:

There are twenty squares which come in 5 different sizes.   There are 14 squares with sides that are perpendicular to the horizontal (see the first three pictures below).   But there are 6 more squares whose sides are not perpendicular to the horizontal (see the last two pictures below).   The last two pictures might provide a good opportunity to discuss why the figures are squares.   Use the fact that perpendicular line segments have slopes which are negative reciprocals of each other.   In the first picture, the slopes are +1 and -1.   In the last picture, the slopes are +2 and - 1/2.

The number of different squares of each size is given below:



John Overton gets EXTRA CREDIT on this problem for sending in a general formula for the number of squares that can be produced from any square matrix of circles of dimension n x n.

Click here for John's excellent analysis of this problem.

Correctly solved by:

1. Sagar Patel	Brookstone School Columbus, Georgia
2. James Alarie	University of Michigan -- Flint Flint, Michigan
3. Richard K. Johnson	La Jolla, California
4. John Funk	Ventura, California
5. John Overton	Saddleback Valley Unified School District Mission Viejo, California
6. Neal Amos	Brookstone School Columbus, Georgia
7. Mr. Robb's Discrete Math Class	John Handley High School Winchester, Virginia
8. Shaan Arora	Brookstone School Columbus, Georgia

Send any comments or questions to: David Pleacher