We discuss the number of proper colorings of hypercubes
given q colors. When q=2, it is easy to see that there are only 2
possible colorings. However, it is already highly nontrivial to figure
out the number of colorings when q=3. Since Galvin (2002) proved the
asymptotics of the number of 3-colorings, the rest cases remained open
so far. In this talk, I will introduce a recent work on the number of
4-colorings, mainly focusing on how entropy can be used in counting.
This is joint work with Jeff Kahn.