AMC 12 Mock Exam 1 Problem 16

Two pots have numbers differing by $3$ only if they are in the same residue class modulo $3$, and the restriction forbids choosing two consecutive members of the same class. The classes are the chains $$1,4,7,10;\qquad 2,5,8,11;\qquad 3,6,9,12,$$ and the choices within different chains are independent.

In a chain of four pots, let $a_n$ be the number of ways to choose from the first $n$ with no two adjacent. Then $a_1=2$, $a_2=3$, and $a_n=a_{n-1}+a_{n-2}$ (either skip pot $n$, or choose it and skip pot $n-1$), so $a_3=5$ and $a_4=8$. The total is $8^3=\boxed{512}$.