## Thursday, March 09, 2006

### Hyperfun!

Smullyan's favourite game is called the Hypergame, invented by Zwicker. I hadn't heard of it until today when I read about it in Barwise and Moss's book Vicious Circles.

Consider games where the players take turns and the game is guaranteed to terminate in a finite number of moves. These are call well-founded games.

The Hypergame is quite simple. The first player chooses a well-founded game. The game now turns into that game and the second player starts by opening in that game with play continuing in that game. Eg. if the first player says "Chess (with whatever house rules are required to make it terminate)" then they start playing Chess with the second player moving first. Because of the stipulation of well-foundedness the Hypergame lasts precisely one move more than some well-founded game. Therefore the Hypergame is itself well-founded and always terminates.

Anyway, Alice and Bob decide to play:

Alice: Let's play the Hypergame!
Bob: Cool! I'll go first. My move is to select the Hypergame.
Alice: OK, now we're playing the hypergame. So my first move is to select the Hypergame.
Bob: The Hypergame is cool, I pick that.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
etc.

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