Question 23
Given an $n$-tuple of numbers $(x_1, x_2, \dots, x_n)$ where each $x_i = +1$ or $-1$, form a new $n$-tuple
$$
(x_1x_2, x_2x_3, x_3x_4, \dots, x_nx_1),
$$
and continue to repeat this operation. Show that if $n = 2^k$ for some integer $k \ge 1$, then after a certain number of repetitions of the operation, we obtain the $n$-tuple
$$
(1, 1, 1, \dots, 1).
$$
$$
(x_1x_2, x_2x_3, x_3x_4, \dots, x_nx_1),
$$
and continue to repeat this operation. Show that if $n = 2^k$ for some integer $k \ge 1$, then after a certain number of repetitions of the operation, we obtain the $n$-tuple
$$
(1, 1, 1, \dots, 1).
$$