Question 3
Let $n$ be a positive integer. For every integer sequence $A=(a_0,a_1,\ldots,a_n)$ satisfying $0\le a_i\le i$, define $t(A)=(t(a_0),t(a_1),\ldots,t(a_n))$ by letting $t(a_i)$ be the number of terms preceding $a_i$ in $A$ that are different from $a_i$. Show that fewer than $n$ applications of $t$ lead from any such $A$ to a sequence $B$ with $t(B)=B$.