Question 77
In each vertex of a regular dodecagon $A_1A_2...A_{12}$ one of the numbers $1$ or $-1$ is written. Initially, number $-1$ is written in $A_1$, and number $1$ is written in all other vertices. It is allowed to simultaneously change the sign of the numbers written in any six consecutive vertices of the dodecagon.
Prove that after repeating this procedure several times it is not possible to have $-1$ written in $A_2$ and $1$ in all other vertices. (Counting and Configurations)
Supplemental illustration (AI-generated):
Prove that after repeating this procedure several times it is not possible to have $-1$ written in $A_2$ and $1$ in all other vertices. (Counting and Configurations)
Supplemental illustration (AI-generated):