Hey @amusingminnow, you're exactly right! Basically, it means that flipping the necklace is still the same necklace.

I like the idea of dividing out the rotations and flips. However, that gives you 1.5, like you showed us. That doesn't make sense, right? We can't have a non-whole number amount of configurations. So, there must be some sort of flaw that we need to find out.

Let's walk through the method: there are \(4!\) arrangements ignoring all conditions. However, since two of them are the same color, we divide by \(2!\). \(\frac{4!}{2!}\) gives us the arrangements ignores the reflection and rotation requirement, as if they were in a line.

Now, the reason why we need to divide is because we overcounted the reflections and rotations of the same exact arrangement. But, here's the problem with just simply dividing by \(4\cdot2\). notice how for some of the arrangements, flipping does the exact same thing as rotating \(180^{\circ}\), like this one:

So not every arrangement is overcounted exactly \(4\cdot 2\) times.

To see how it really works, let's look at all of the \(\frac{4!}{2!}\) arrangements.

(a little pink, oops )

Can you see how all of the arrangements in group 1 are the same, and all the ones in group 2 the same? We can't just simply divide by \(8\), because not every arrangement is overcounted \(8\) times.

The main difference between the groups is that in one, the two beads with the same color are touching and in the other, they are not, which is why the given solution works. Hope this clears everything up!