MathDB

In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students

A_1, A_2, \dots, A_{2023}

is called mischievous if

\bullet

Total number of friends of

A_1

is odd.

\bullet

A_i

and

A_{i+1}

are friends for

i=1, 2, \dots, 2022

\bullet

Total number of friends of

A_{2023}

is even.
Prove that the total number of mischievous sequences is even.

Problem Statement