Multiplayer Sudoku

Arnold Kling suggests creating Sudoku for two.

Instead of numbers, use colors. It looks nicer, and I think it would be more fun.

Have an electronic board that generates the puzzles and keeps track of each player's time. When the game starts, it's the first player's turn. His clock is running. He selects a color and places it on the board (by pushing on the spot--the electronic board will sense both the color he picked and the place he marked). If his placement is correct, it is the other player's turn. The other player's clock starts, and he selects a color and places it on the board. Play continues until one player runs out of time (loss) or makes an incorrect move. If the puzzle is solved with neither player losing, it is a draw.

I like this idea, except that it's not very interactive. It's essentially a race: who can solve a Sudoku puzzle faster? I can imagine a much more interactive Sudoku game for two or more players. Moreover, no electronics are required.

Each player gets a sheet of paper with a completed Sudoku on it. Each player keeps his Sudoku hidden from his opponents. Using pencils (or numbered/colored tiles), the players take turns filling in cells of a common Sudoku grid, which is initially blank. Once the common Sudoku grid is full, each player gets a point for each cell of his private Sudoku that is in agreement with the common grid. The highest scorer(s) is the winner(s).

To make the game nontrivial, players are not allowed to fill in a cell arbitrarily. It is not legal to fill in a cell if it would result in a row, column, or block containing two of the same number. Thus, the common grid will be a valid Sudoku once it is completely filled in. If a player is unable or unwilling to fill in a cell, then he may pass. However, if all players pass in succession, then the next player must either fill in a cell or erase a cell.

Obviously, there is no guarantee the common Sudoku grid will be completely filled. Therefore, players should either agree to a time limit or a limit on the number of turns. Also, it would be a good idea to encourage players not to pass unnecessarily. If a player fills in a cell after the previous player passed, then he should get a scoring bonus. My initial guess is that this bonus should be one point, but a more desirable bonus value (not necessarily a whole number) might be found experimentally.

Update: The first player presumably has a small advantage, so the game should be played repeatedly such that each player goes first the same number of times.