This is a variety of Golomb's game of pentominoes. A pentomino is a shape made by conjoining five squares in any one of 12 possible configurations (18 if you distinguish reflections as well as rotations). In Golomb's game you each in turn place a pentomino either way up and any way round on any five contiguous squares of an 8x8 grid such as a chessboard until one of you cannot add another, thereby losing. (There is a known win for the first player, but no one can remember it.) I have always loved this game - in fact, my first contribution to Games & Puzzles magazine, way back in 1974, was a whole series on Golomb's game and related puzzles.
In order to play it you need, of course, to make a set of pentominoes of the right-size individual squares. You can do so using a scalpel (carefully) or craft knife based on the template illustrated (to come).
Play. Each of you adopts two personal colours without duplication. You then play pentominoes in the usual way. In this case, however, your aim is to cover as many squares of your opponent's colours as possible while leaving a greater number of your own uncovered.
Score. You each score the number of one of your colours still visible by the number of those of your other colour. Consider the illustrated example. The number of yellows uncovered is 7, of reds 5, blues 4, greens 3. If you're playing red/blue, you win by 1 point, with 21 to 20. Yellow/blue gets you 13 points, with 28 points to 15. And red/yellow by 23 points, with 35 to 12. Note, by the way, that the five empty squares in the middle can be filled by just one pentomino, but it's already in use at top right.