Reversi (Othello) is played with black & white pieces on a board with 64 spaces. I won’t waste your time describing the game, if you know the answer to this question you won’t need an explanation anyway.
To begin the game there are very few moves that can possibly be made. I’m no expert, but there are exactly two different ones? Four different ones? In any event, there is an extremely limited, finite number that can possibly be made as the “first” move.
After that first opening more there are a larger number of “next” possible moves, yet relatively few and certainly a finite number. After that move, again there are more, but still finite, possible moves. As so on and so on.
Given that after each move the number of possible “next moves” is finite (albeit getting larger as the game goes on), I would assume there is a finite number of possible games in their entirety.
How many?
(I contrast this with chess, where, theoretically, there are an infinite number of *possible *moves, if the players just keep chasing each other around the board.)
Do I have my thinking straight on this? Am I missing something?
Chess does not have an infinite number of possible moves (or positions for that matter).
If neither player has made a capture or a pawn move for 50 moves (by each side), the player to move can claim a draw.
There are 4 possible opening moves in Othello, but you have to remember there is a strong element of symmetry (in a game with a square board and only two pieces).
Off the top off my head, there are only 3 possible replies to the first move.
The players collectively place 60 stones (64 squares less the initial four) and if there is an average of six possible moves per turn, I figure the number is finite but so huge as to be beyond full exploration. Wiki says a 6x6 grid is solved, but the conventional 8x8 is not.
There isn’t a requirement to claim a draw after 50 moves with no capture or pawn move, so it could go on for a very long time if 2 very stubborn players with nothing better to do could be found.
Here’s an upper limit: there are sixty possible moves on the first turn. (Not true, of course, but there are certainly not MORE than sixty, because there are only sixty available squares.) The second turn cannot have more than fifty-nine moves, and so on… which makes the total possible 60! does it not?
60! is an upper bound, but the actual number of possible games is probably significantly lower. As you noted, not every move is legal, and so many of the games you’ve counted here can’t actually happen.
Since each piece placed must touch another piece you can reduce the upper limit drastically below below 60! There are 12 squares touching the first four stones so there are at most 12 first moves. (I know there are many fewer). Each stone that is added removes at one available square and adds at most 7. So an upper bound on starting moves would be
12192633404754
at this point there are only 53 squares left so an upper bound is