−1. Thus the surreal number notation is {1|−1}. However, this is not a surreal number because 1 > −1. This is a Game but not a number. The notation for this is ±1, and it is a hot game, because each player wants to move here. This is a 2×3 grid, which is even more complex, but, just like any Domineering game, it can be broken down by looking at what the various moves for Left and Right are. Left can take the left column (or, equivalently, the right column) and move to ±1, but it is clearly a better idea to split the middle, leaving two separate games, each worth +1. Thus Left's best move is to +2. Right has four "different" moves, but they all leave the following shape in some rotation: This game is not a hot game (also called a cold game), because each move hurts the player making it, as we can see by examining the moves. Left can move to −1, Right can move to 0 or +1. Thus this game is {−1|0,1} = {−1|0} = −½. Our 2×3 grid, then, is {2|−½}, which can also be represented by the mean value, ¾, together with the bonus for moving (the "temperature"), 1¼, thus: High-level play The Mathematical Sciences Research Institute held a Domineering tournament, with a $500 prize for the winner. This game was played on an 8×8 board. The winner was mathematician Dan Calistrate, who defeated David Wolfe in the final. The tournament was detailed in Richard J. Nowakowski's Games of No Chance (p. 85). Winning strategy A problem about Domineering is to compute the winning strategy for large boards, and particularly square boards. In 2000, Dennis Breuker, Jos Uiterwijk and Jaap van den Herik computed and published the solution for the 8x8 board. The 9x9 board followed soon after some improvements of their program. Then, in 2002, Nathan Bullock solved the 10x10 board, as part of his thesis on Domineering. The 11x11 board has been solved by Jos Uiterwijk in 2016. Domineering is a first-player win for the 2x2, 3x3, 4x4, 6x6, 7x7, 8x8, 9x9, 10x10, and 11x11 square boards, and a second-player win for the 1x1 and 5x5 boards. Some other known values for rectangular boards can be found on the site of Nathan Bullock. Cram Cram is the impartial version of Domineering. The only difference in the rules is that