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First, I imagined how you could use the most number of tiles up using more than two of the same colour; you would follow it all the way around, clockwise. Then I decided to try two rows at a time for example: You start at the top right orange corner. Go right 1 then down 1, right 1 and up 1. Continue until you reach the corner. Enter the corner diagonally then 1 across and exit diagonally as demonstrated in the picture when you have gone the whole way round enter the next rows diagonally and repeat, this time three rows at a time. When there is only three squares left, enter straight into the middle.