Ulam's Problem Revisited
Is it possible for a question to be almost resolved? Below, a matter that is all but put to rest. We present it here partly in the wild hope of having the solution finished (we suspect that may be beyond human capability) and partly with the idea of giving as widespread publicity as possible to an intriguing problem that has fascinated many (we have not seen it published in a very long time).
In A Collection of Mathematical Problems, S. M. Ulam asked (and in a later edition announced the answer to) the question of whether there exists a deal on which South can make seven of any suit against any defense but cannot make six notrump. The answer is yes. We'll leave you the fun of finding a solution for yourself. Now to our current matter:
Question 4: Does there exist a deal on which South can make seven of any suit against any defense but cannot make five notrump?
Why do we say this question is "almost" resolved? People who have worked on it are convinced that the answer is no. A few have developed proofs that it is impossible to construct such a deal on the "obvious" pattern, which we will describe (though not define) as the kind of deal that leads to known positive solutions for the case of six notrump. But how, if at all, can one develop a more general proof? If we can't, will we ever "know" the answer?
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