A white equilateral triangle T with side length 2022 is divided into equilateral triangles with side 1 (cells) by lines parallel to the sides of T. We'll call two cells <spanclass=′latex−italic′>adjacent</span> if they have a common vertex. Ivan colours some of the cells in black. Without knowing which cells are black, Peter chooses a set S of cells and Ivan tells him the parity of the number of black cells in S. After knowing this, Peter is able to determine the parity of the number of <spanclass=′latex−italic′>adjacent</span> cells of different colours. Find all possible cardinalities of S such that this is always possible independent of how Ivan chooses to colour the cells. combinatoricsParitycolouring