Question 109
An equilateral triangle $T$ with side length $2022$ is coloured in white and partitioned into equilateral triangles with side length $1$ (called *cells*) through lines, parallel to the sides of $T$. Two cells are called *adjacent* if they have at least one vertex in common. Ivan colors some of the cells in black. Without seeing which cells are black, Peter selects only once a set $S$ of cells (containing at least one) and asks Ivan if the number of blacks among the selected ones is even or odd. After receiving an answer, Peter is able to tell if the number of pairs of differently colored adjacent cells in $T$ is even or odd. Find all possible values of the number of elements of $S$ if that is always possible no matter the coloring of Ivan.