Question 67
Let $n \neq 0$. For every sequence of integers
$$
a = a_0, a_1, a_2, \dots, a_n
$$
satisfying $0 \le a_i \le i$, for $i = 0, \dots, n$, define another sequence
$$
t(a) = t(a)_0, t(a)_1, t(a)_2, \dots, t(a)_n
$$
by setting $t(a)_i$ to be the number of terms in the sequence $a$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $a$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $b$ such that $t(b) = b$.
$$
a = a_0, a_1, a_2, \dots, a_n
$$
satisfying $0 \le a_i \le i$, for $i = 0, \dots, n$, define another sequence
$$
t(a) = t(a)_0, t(a)_1, t(a)_2, \dots, t(a)_n
$$
by setting $t(a)_i$ to be the number of terms in the sequence $a$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $a$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $b$ such that $t(b) = b$.