n children are playing on a carousel with n seats.
How many ways are there to change the sitting order, such that no child is sitting in front of the child they are sitting in front of now?
It can be something like number of permutations with one fixed point or Hat-Check Problem, but I am not sure how to solve it. Thanks for any help.
The first problem with your question is you are assuming $n$ is even, isn’t it? If is odd you are siting a children in front of two halves of another kids and my answer require you cut them in halves, I'm really sorry for them.
This is a hint:
Consider $n=2^k$ and try this: if $n=2m$ sit $m$ fixed (by nails or adhesive tape I suposed) and rotate the other $m$ so each fixed have seen the other $m$. Take 2 carrousels with $m$ sits and tit the first $m$ in one and the others in the other and repeat the process (notice that in this case you use $m/2$ move). Go now with 4 and so… Sum up all moves: $$n/2+n/4+…+1=\sum_{i=0}^{k-1}2^i=n-1.$$
If $n$ is not a power of two you should left a kid out in some point, that’s a sad thing.