Abstract
Afamous, well-traveled, and relatively easy counting problem is that of determining the number of squares of aU sizes on a checkerboard. In this article, we pose and solve several related extensions involving enumerating squares and rectangles. Preservice and in-service secondary teachers participating in a problem-solving course at the University of Northern Colorado produced most of the investigations and conclusions illustrated here. In addition, a recent statewide mathematics contest, attended by about a thousand students in grades 7–12, included versions of the explorations. We were pleased that a significant number of students successfully rose to the challenge. Given appropriate guidance, these problems can be a source for wonderful classroom explorations and discovery for both the student and teacher.
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