Problem:
The nine horizontal and nine vertical lines on an checkerboard form rectangles, of which are squares. The number can be written in the form , where and are relatively prime positive integers. Find .
Solution:
Notice that rectangles must be formed by choosing two distinct vertical lines and two distinct horizontal lines. Because there are nine vertical lines and nine horizontal
lines, the total number of rectangles is . Of these, are squares, are squares, and, in general, are squares, because each square is determined by its size and the position of its upper-left corner. Hence the total number of squares on the checkerboard is . Thus
and .
The problems on this page are the property of the MAA's American Mathematics Competitions