
Re: A rational X irrational game
Posted:
Mar 19, 2004 7:57 PM


Keith A. Lewis wrote: > artur@opendf.com.br (Artur) writes in article <93b1dfbf.0403190730.4e9cc2f8@posting.google.com> dated 19 Mar 2004 07:30:09 0800: > >>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 the players agree and cooperate to converge on a single number, you can > pin it down like that. But they can't be guaranteed to do that (in fact > they probably won't if they are competitive). >
How can the final intersection of these nested intervals, no matter how they are chosen, ever contain more than a single number? The nth interval is shorter than 1/n in length. If there are two numbers in the intersection, they differ by some amount, and there is a N so that the reciprocal 1/N is smaller than that difference. That means that from step N on, at most one of those numbers is in the intersection. The intersection must be nonempty, since these are closed intervals of R, and the real line has the property that a nested sequence of nonempty compact intervals has nonempty intersection.
> >>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. > > > Within any In of length Ln > 0, there will be both rational and irrational > numbers. In fact, the cardinality of the respective sets of remaining > rational and irrational numbers after move n is the same as it was before > move 1. The game never ends. >
It ends in the limit. Take 1/2 second for step 1, 1/4 second for step 2, and so forth with 1/2^N second for step N. The game is over in one second, albeit the moves are somewhat rapid towards the end.
> >>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. > > > Keith Lewis klewis {at} mitre.org > The above may not (yet) represent the opinions of my employer.
Dale

