[This is post #21 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]
Knowing the color pattern around any one of the twelve faces (5-sided stars) of a Straws Thingy tells us exactly how the tetrahedra are colored, so it lets us reconstruct the entire color scheme. In other words, any one face determines all 12 faces. This is useful for quickly comparing color schemes. The “original” Straws Thingy from yesterday has these twelve faces, in no particular order:
For example, the face pictured in the top left has the colors Blue (B), Orange (O), Pink (P), Green (G), and Yellow (Y) in counterclockwise order around the face, so we’ll write it as BOPGY. It also doesn’t matter where we start—this is the same ordering as PGYBO, or YBOPG, or…—so to be consistent, let’s choose to always start at Blue. Then these twelve orderings are, respectively:
BOPGY, BPYGO, BGYOP, BPGOY, BGPYO, BYPGO, BOYPG, BYOGP, BPOYG, BOGYP, BYGPO, and BGOPY.
You can check that all 12 of these orderings are different, but remember that any one of them can be used to identify this original Straws Thingy.
So if we’re handed a new Straws Thingy and we want to know if it matches the original one, we only need to check a single face of the new one: if it’s in this list, the two Straws Thingys are identical, so the twelve faces of the new one will be exactly these twelve faces. If not, then there can be no face matches at all: the twelve orderings of the new faces can’t be in the above list.
So how many orderings aren’t in this list?
Well, how many orderings are there in total? We have to start with Blue, but then any of the four remaining colors can come next, then any of the three colors can come after that, then either of the two remaining colors follows, finishing with the only color remaining. So there are \(4\times 3\times 2\times 1 = 4! = 24\) face orderings in total.[1]
With 12 of those orderings belonging to the “original” Straws Thingy, there are only 12 other faces to pick from. So any other Straws Thingy, like yesterday’s “odd one out,” has to use exactly the other 12 orderings. In other words, there’s only one other Straws Thingy color arrangement! Any Straws Thingy you build will match either the “original” or the “odd one out”; which did you build?
Notes
- That middle term is not an excited “4”. It’s pronounced 4 factorial, and in general, \(n! = n\times (n-1)\times\cdots\times 2\times 1\) counts the number of ways to arrange n items in a line, or as we saw here, \(n+1\) items in a circle. [↩]