MathDB

Consider a sequence of numbers

(a_1, a_2, \ldots , a_{2^n}).

Define the operation

S\biggl((a_1, a_2, \ldots , a_{2^n})\biggr) = (a_1a_2, a_2a_3, \ldots , a_{2^{n-1}a_{2^n}, a_{2^n}a_1).}

Prove that whatever the sequence

(a_1, a_2, \ldots , a_{2^n})

is, with

a_i \in \{-1, 1\}

for

i = 1, 2, \ldots , 2^n,

after finitely many applications of the operation we get the sequence

(1, 1, \ldots, 1).

Problem Statement