We start off picking hole 2. Assume we don’t get it. That means on the second day it could be in any of the holes except 1. Now we check hole 3. Assume we don’t get it. So on the second day it’s in hole 2, 4, or 5. That means that on the third day the only even hole it could be in is 4. So we check 4. Assume we don’t get it. Then it must have been in hole 2 or 4 on the second day, so on the fourth day it is again in hole 2 or 4.

Now we check hole 2. If we didn’t get it, it must be in hole 3 or 5. So we check 3, and now if we didn’t get it must be in hole 4, so we check 4 and we are certain to get it.

So the final sequence is 2, 3, 4, 2, 3, 4. ]]>

[spoiler]

Check hole #2

Uncaught fox could be in 1, 3, 4, 5

After moving it could be in 2, 3, 4, 5

Check hole #3

Uncaught fox could be in 2, 4, 5

After moving it could be in 1, 3, 4, 5

Check hole #4

Uncaught fox could be in 1, 3, 5

After moving it could be in 2, 4

Check hole #4

Uncaught fox will be in 2

After moving it could be in 1, 3

Check hole #3

Uncaught fox will be in 1

After moving it will be in 2

Check hole #2

There are other patterns that would work like 234234 and reflections like 432432[/spoiler]

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