Today I encountered a tricky counting problem: ‘There are necklaces which consist of four white beads and four black beads on a string. How many such necklaces are distinguishable?’.

This sounds like a simple problem. I immediately thought the answer was eight choose four divided by eight divided by two. Eight choose four relative positionings of the black and white beads, divided by eight to indistinguish rotations, divided by two to indistinguish flipping.

Seems right, huh? That number is not an integer! Oh no! How can this be? One part of the trick is that rotations and flips ‘interact’ in certain cases. Here is an example: consider two necklaces which include the bead sequences BBWBWBWW and BBBBWWWW. There are sixteen (indistinguishable) ways to position the first one: eight rotations times two flips. However, there are less than sixteen ways to position the second.

I manually enumerated the indistinguishable necklaces and found that there are eight of them. Unfortunately, I haven't been able to find a combinatoric justification for this number. If you can think of a solution tell me tell me tell me!