I hope it’s answer should be 20.
Probably can be solved using Theory of Magic squares
see these links to explore.. how Magic squares are created
https://www.math.hmc.edu/funfacts/ffiles/10001.4-8.shtml

this one to see.. that Each magic sqaure can produce 8 distinct magic squares
https://en.wikipedia.org/wiki/Magic_square

7 days..
He can go in clock and anti clockwise together.. lets say he check… 7 and 8 one day.. then 6 and 9 ..then 5 and 10 ..then 4 and 11 then 3 and 12 …. 2 and 13 .. either he would encounter in any one of these cases.. worst sceneario is when thief is found in the last attempt of checking in cave 1 (left still to check) .. and hence 7 days are sufficient to determine.
If any loophole is there…i welcome the suggestion