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Topic: A rational X irrational game
Replies: 10   Last Post: Mar 20, 2004 5:52 PM

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Fred Galvin

Posts: 1,758
Registered: 12/6/04
Re: A rational X irrational game
Posted: Mar 20, 2004 2:31 PM
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On 19 Mar 2004, Artur wrote:

> Hello
> I've been thinking about the following problem (or puzzle) but hasn't
> come to a conclusion yet. Maybe someone can give a hint.
> A and B are playing the following game: On the real line, A chooses a
> closed interval I1 of length 0<L1<=1. Then, B chooses a closed
> interval I2, of length 0<L2<=1/2, contained in I1. Then A, in turn,
> chooses a closed interval I3, of length 0<L3<=1/3, contained in I2,
> and so on. We know there's one, and only one, real number x that
> belongs to all of the intervals I1, I2, I3.... If x is rational, A
> wins the game, and if x is irrational, then B wins. We are asked to
> find an strategy that B should follow so that he will certainly win
> the game no matter how B chooses his intervals.

If you generalize, replacing "x is rational" and "x is irrational" by
"x belongs to E" and "x does not belong to E", where E is a given set,
this is (essentially) Mazur's game, also known as the Banach-Mazur
game. (Mazur invented the game and made a conjecture, which Banach
proved.) Such games were discussed by the Polish mathematicians Mazur,
Banach, and Ulam in 1935, in the famous Scottish cafe. By now there is
a vast literature about those games, which have been much generalized.
Here are a couple of references.

R. Daniel Mauldin (editor), _The Scottish Book_, Birkhauser, 1981,
ISBN 3-7643-3045-7. (See Problem 43.)

Rastislav Telgarsky, "Topological games: on the 50th anniversary of
the Banach-Mazur game", Rocky Mountain J. Math. 17 (1987), 227-276.

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