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

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: A rational X irrational game
Posted: Mar 19, 2004 10:49 AM
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On 19 Mar 2004 07:30:09 -0800, (Artur) wrote:

>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.
>Since the rationals are countable and the irrationals are not, it
>really seems that there is such a strategy that assures B will ever
>win, but I couldn't find it so far.

Hint: B doesn't have to force a win on the first move. On
the first move B prevents one way he can lose, on the
next move he prevents another method of losing...

>Thank you.


David C. Ullrich

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