Kovanchi A.K.

At each vertex of a cube, a number is written. Instead of each number, the arithmetic mean of the numbers in the three adjacent vertices is written (the numbers are replaced simultaneously). After ten such operations, the original number reappeared at each vertex. Were the original numbers necessarily the same?
We will color the vertices in black and white so that the ends of each edge are of different colors. We will write ones in the white vertices and zeros in the black ones. After each pair of operations, the arrangement of numbers repeats, meaning that after ten operations - as well.

## Answer

Not necessarily.

## Problem