As many of you might have seen before, here is the description of the classic weighing balls problem:
One of twelve pool balls is a bit lighter or heavier (you do not know) than the others. At least how many times do you have to use an old balance-type pair of scales to identify
this ball and tell if it is heavier or lighter?
I know the answer is $3$, which is the lower bound given by the information theory since $\lceil\log_3(2\cdot12)\rceil=3$. Now my question is: can the answer still remain be $3$ if the number of balls is $13$ rather than $12$ (since $\lceil\log_3(2\cdot13)\rceil=3$ as well)? More general, how does the parity of the number of balls here affects the answer in our weighing ball context?