In [From knights to kings](https://puzzling.stackexchange.com/questions/3337/from-knights-to-kings), we were asked

> 
> On a NxN chess board $N^2$ knights are placed (one per cell). Each pair of knights, who control each other (i.e. one move away from each other) are friends.
>  One day all knights are promoted to kings. Is it possible to put them on the board (in different order) that all friends will be similarly one move away from each other?
> 
> 
> 

We found out that for a square board, we can't put all the knights next to their friends when $N>3$. What if we are using a $N\times M$ board ($N\ge M$) instead? For example, if $N=3,M=2$:

```
A B C
D E F

```

You only need to swap D and F to get a solution

```
A B C
F E D

```

It's fairly obvious that all $N\times 1$ and $N\times 2$ boards will work. What other board sizes will work under these conditions?