Question 97
Let $n$ be a positive integer and let $W=\cdots x_{-1}x_0x_1x_2\cdots$ be a periodic two-sided infinite word, with each $x_i$ equal to $a$ or $b$. Its least period $p$ exceeds $2^n$: $x_i=x_{i+p}$ for every integer $i$, and no smaller positive period works. A nonempty substring $U$ of length at most $p$ is called FU if all four words $aU,bU,Ua,Ub$ occur in $W$. Prove that $W$ contains at least $n$ distinct FU substrings.