A 4 by 4 square is divided into sixteen unit cells. Each unit cell is coloured with one of four available colours, red, blue, green or yellow. The 4 by 4 square contains nine different 2 by 2 "sub-squares". Suppose that we colour the sixteen unit cells in such a way that each 2 by 2 sub-square has one cell of each colour. Prove that the four corner cells in the large 4 by 4 square must then be coloured differently.
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