Question 78
Let $S=\{1,2,4,8,16,32,64,128,256\}$. A subset $P$ of $S$ is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set $Q$ is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets.
(A set $A$ is said to be a proper subset of a set $B$ if $A$ is a subset of $B$ and $A \neq B$.)
(A set $A$ is said to be a proper subset of a set $B$ if $A$ is a subset of $B$ and $A \neq B$.)