Question

Eleanor is going to color the sides of a regular octagon. Two sides will be red, two sides will be yellow, two sides will be green, and two sides will be blue. Furthermore, opposite sides of the regular octagon will have the same color. How many different ways can Eleanor color the sides of the ocatgon? (Two patterns are considered identical if one can be rotated to form the other.)

Answer 1

Answer: There are 12 ways to color the octagon

Explanation:

Red-Yellow-Green-Blue-Red-Yellow-Green-Blue: This is the simplest way to color the octagon. The first two sides are red, the next two sides are yellow, and so on.

Yellow-Red-Green-Blue-Yellow-Red-Green-Blue: This is the other way to color the octagon. The first two sides are yellow, the next two sides are red, and so on.

In each case, there are 4! = 24 ways to order the colors.

However, we need to divide by 2 because the two patterns are considered identical if one can be rotated to form the other.

Therefore, there are 24 / 2 = 12 different ways to color the sides of the octagon.