A Symmetrical Octagon

Problem 6 – Correct! – Score: 1 / 7 (27650)

Report an ErrorProblem:

Ellie wants to color two sides of a regular octagon green, two blue, two red, and two yellow such that each pair of opposite sides of the octagon have the same color. How many different patterns can she form? (Two patterns are considered identical if one can be rotated to form the other.)

Solution:

Suppose she first chooses a side to color green. Because two patterns are considered identical if one can be rotated to form the other, it doesn't matter which side she chooses. She can then spin her octagon so that this first colored side is on "top". Then, suppose she goes clockwise around the octagon to finish coloring. She can't use green because the other green side must be opposite the first side. So, she has  choices for the first side after the green side. She then has  colors remaining for the next side, and remaining for the side after that. At this point, she has four consecutive sides that are four different colors. The other four sides are then determined: each is the color of its opposite side. So, there are  possible colorings.

Hint(s):

Grab some crayons and try it!

Your Response(s):

• :( 2520

• :( 315

• :( 3

• :( 24

• :) 6
• Analysis: There is only 4 sides that matter, since the other four sides are mirrored. There are 4! ways to color the four sides, but we overcounted because of rotations. 4!/4 = 3! = 6.

Symmetry, Reflection, and When to Divide

  Level 25 Counting & Probability  +123 XP

Review

In topic: Counting with Symmetry (Counting & Probability).

 

Incorrect. Oops!

In how many ways can  people sit around a round table if no two of the  people Pierre, Rosa, and Thomas can sit next to each other?

Your First Answer: 384

Your Second Answer: 1008

Solution:

After Pierre sits, we can place Rosa either two seats from Pierre (that is, with one seat between them) or three seats from Pierre. We tackle these two cases separately:

Case 1: Rosa is two seats from Pierre. There are  such seats. For either of these, there are then four empty seats in a row, and one empty seat between Rosa and Pierre. Thomas can sit in either of the middle two of the four empty seats in a row. So, there are  ways to seat Rosa and Thomas in this case. There are then  seats left, which the others can take in  ways. So, there are seatings in this case.

Case 2: Rosa is three seats from Pierre (that is, there are  seats between them). There are  such seats. Thomas can't sit in either of the seats directly between them, but after Rosa sits, there are  empty seats in a row still, and Thomas can only sit in the middle seat of these three. Once again, there are  empty seats remaining, and the  remaining people can sit in them in  ways. So, we have seatings in this case.

Putting our two cases together gives a total of  seatings.