Question 6
Let $n$ be a positive integer. We call *smooth* a sequence of integers $a_1, a_2, \dots, a_k$, with $1 \le a_i \le n$, if there exists an integer $m$, with $1 \le m \lt k$, such that $a_1 = a_{k-m+1}, a_2 = a_{k-m+2}, \dots, a_m = a_k$. Furthermore a sequence is *universal* if each of the sequences obtained through replacing $a_k$ by $1, 2, \dots, n$ is smooth. For each $n$ find a universal sequence of minimum length.