KöMaL 2018 B4986

Consider the 64 points of the space for which each of the three coordinates is 1, 2, 3 or 4. Kate and Peter are playing a three-dimensional tic-tac-toe game on this set of points. Kate starts the game by selecting any point and colouring it blue. In the second step, Peter selects a different point and colours it red. Then they take turns by selecting further points and colouring them in blue or red. Whoever first completes a collinear set of four points of their own colour will win the game. Show that it makes no difference for Kate whether she starts by colouring the point $(1,1,2)$ or the point $(2,2,1)$ blue in the first step.