Original Question: Students sit at their desks in three rows of eight. Felix, the class pet, must be passed to each student exactly once, starting with Alex in one corner and finishing with Bryn in the opposite corner. Each student can pass only to the immediate neighbour left, right, in front or behind. One possible path is shown. How many different paths can Felix take from Alex to Bryn?
So what confuses me about this question is typically this style of "how many paths" gives a more limited direction, e.g. only right and up towards the target. However, in this question, the students can pass left, right, in front, and behind. So the traditional method I usually use for these type of questions of counts on points wouldn't work (finding sum of number of ways to each point)? Or at least I think it doesn't work.
Best Answer
The answer is $2^{c-2}$ where $c$ is the number of columns. The trick is to notice the distribution of unconnected columns on the top and bottom row.
They are also visible in the middle row.