FMM12

Friday Morning Math Problem

Checkerboard Color Schemes

Two of the squares of a 7 x 7 checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?

Hint: There are ${\binom {49}{2}}=1176$ ways to select the positions of the yellow squares. However, because we can apply quarter-turns, there are less than 1176 inequivalent color schemes.

Solution

Color schemes fall into two classes: 1. color schemes in which the two yellow squares are not diametrically opposed 2. color schemes in which the two yellow squares are diametrically opposed Case (1) appears in four equivalent forms, so we will divide the total number of color schemes in Case (1) by 4. Case (2) appears in two equivalent forms,so we will divide the total number of color schemes in Case (2) by 2. We also know there are ${\dfrac {49-1}{2}}=24$ such pairs of yellow squares, contributing (24/2) total inequivalent color schemes. The number of cases in the Case (1) class is the total number of arrangements, minus the 24 pairs of yellow squares in the Case (2) class - and we divide it by 4, so Case (1) contributes (1176 - 24)/4 total inequivalent color schemes. The total is therefore: ${\dfrac {1176-24}{4}}+{\dfrac {24}{2}}=300$