> 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.