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We have not played Arguile enough times to give a firm sense of whether we are employing rational strategies or not, but in limited play, I have found the game to be very elegant, and I like it at least as much as other abstract space-relations strategy games in my possession such as Pipeline and Quoridor.
Each side lines up checker-like disks on a series of spaces in a row on their side of the board, and they move their pieces diagonally, in an attempt to occupy all four spaces of a diamond at the board's center. There are three such diamonds, and the winner is the person who is able to completely fill in two of them.
Movement is somewhat like checkers, and one can jump one's opponent's pieces, and turn them to his own color.
The interesting variable in the game is that each person is secretly holding three cards that enable one to turn over all of his opponent's disks in a diagonal row. So clearly, a great amount of the strategy is knowing when to pull out these cards. (The three cards one holds are different every game.) If you do it too early, then your opponent has the advantage of saving his cards until the last, optimal moment. If you do it too late, however, an opponent might be able to flip a group of your disks right before moving to complete a diamond--and once a diamond is fully occupied, the disks in it are immune to 're-flipping.'
On the surface, the game seems to have the potential to have duplicative play--there are only so many different sets of three cards that one can hold each game, and the board is small and two opponents might, it seems, repeat the same strategy under the same conditions. But this is only a suspicion of mine--we have not played to that point, and thus far Arguile has proved fresh and interesting with each play.
Checkers is not a bad comparison for one element of Arguile--checkers is an underrated abstract strategy problem because so many people master the rules in childhood. And Arguile marries those virtues with other strategic elements, such as when to play a card that allows for the situation on the board to change drastically.
I am still playing with the theory that it's best to be the last person to unveil his cards, and so far this theory is being sustained. One thing is certain, which is that once you get to the point where you can, via holding the last cards, ensure that your opponent won't have enough disks left to complete two diamonds, you'll eventually win barring a huge blunder. But it's not so easy to ensure that you get in that position.
I must say--I find this game very attractive, very elegant. The only worry I have is that we might 'play it out' before too many plays. But there's no sign of that yet, and I suspect that it would give others a lot of entertainment.