# Riddle of the fox and the holes, with n holes

I was wondering if the problem of fox and holes

(There are five holes in a line. One of them is occupied by a fox. Each night, the fox moves to a neighboring hole, either to the left or to the right. Each morning, you get to inspect a hole of your choice. What strategy would ensure that the fox is eventually caught?)

you can find detailled information here (https://gurmeet.net/puzzles/fox-in-a-hole/, https://www.youtube.com/watch?v=0Prp9n7XfP8)

can be solved for an arbitrary number bigger than 5?

I've solved the case n= {1,2,3,4,5}, but for case 6 it was complicated... do you have any idea ? Maybe it is possible for n odd and not if n even. If it isn't, I'm also searching for an argument of why it isn't :)

thank you!

There is an easy strategy for any $$n$$.
Consider what happens if you inspect the holes $$(2,\ 3,\ 4,\ ...,\ n-1)$$ on successive days. Prove that if the fox starts at an even-numbered hole, then it will not be able to get past your sweep, and that you will therefore catch it.
Therefore the sequence $$(2,\ 3,\ 4,\ ...,\ n-1)$$ followed by $$(n-1,\ n-2,\ ...,\ 4,\ 3,\ 2)$$ will catch the fox.