I've had an idea for a two player sudoku game. it's quite simple, so it wouldn't surprise me if someone else has had a similar idea. The basic idea is you start with an empty grid and take it in turns to fill in an empty cell. however whenever you fill in a cell you also fill in any cells that are set by filling in that cell ie you complete as much of the puzzle as you can given the current state of the grid. You get one point for every cell you fill in. The game finishes when the grid is complete or you cause a contradiction. the winner is the person with the most points.

Now I haven't played this game yet, but the idea of stopping the game by a contradiction is a little worrying. generally it may be quite easy to cause a contradiction, so if you are one or two points ahead it may be easy to win. I think the best way around this problem is to say that if it's possible to fill in a cell without causing an immediate contradiction then you are not allowed to fill in a cell in a way that causes a contradiction.

I think this game would be very interesting if the players agreed on a symmetry for the puzzle at the start of the game. This could be a quarter turn, two reflections (horizontal and vertical or both diagonals) or the whole symmetry of the square. I think to choose the other symmetries would not be so interesting. Then when the first player places his clue the next player is restricted to placing his clue in a cell that is symetrically equivalent to that. This way to play first has a slight advantage. However you can give up that advantage by playing in the central square. It may be advantageous to do this if you get many points for playing there. also it may be possible for the second player to steal the first play advantage. For instance if there are 3 cells that could be played in on a given go, it may be possible to place a clue that fills two of them.

if you play this game with two different coloured pens, say a red one for placing the first clue and a blue one for filling in the consequences, then at the end of the game, if you complete the grid, then the red clues will be a valid sudoku puzzle.