Mathematical problems and games
S. M. Ulam was known for, among other things, the formulation of imaginative problems in a great variety of topics in mathematics and allied areas. His book ``A collection of mathematical problems'' was published in 1960 (see Zbl 0086.241). The problems continued to flow, often in discussions with Paul Erdős, R. L. Graham, R. D. Mauldin and others. Some years ago he invited Mauldin to help him develop, organize and present a new collection of problems, and the present paper is the result of that collaboration. Ulam died in 1984, and Mauldin completed the project, with some assistance from Françoise Ulam, Erdős, Graham, C. C. Rota and others. The material is divided into seven chapters with headings as follows: I. Logic, Set Theory. II. Combinatorics, Game Theory, Number Theory, Complexity. III. Algebra. IV. Topology. V. Real Variables, Functional Analysis, Measure Theory. VI. Metric Spaces, Geometry. VII. Physics and Biology. The chapters are further divided into a total of 79 sections. Along with the problems are comments on progress toward solutions, related known results, bounties offered and the like. A few samples: II.3. For a given subset E of [0,1] players and A and B give in turn a digit from \{0,1\(\}\). Player A wins if the number whose binary expansion results is in E. Consider the same game but with digits chosen from \{0,1,2\(\}\) and ternary expansion. Find a set E which is a winning set for the first player in the binary version but not in the ternary one. II.22.6 Show that for infinitely many n, g.c.d.\(({2n\choose n},105)=1\). III.5. Is there a group structure on a countable set such that the group of endomorphisms is maximal? IV.3. Let E and F be topological spaces and T a mapping from E to F such that if A,B\(\subset E\) are homeomorphic, then T(A) and T(B) are homeomorphic in F. Must T be a homeomorphism? V.3. Is there a measure m defined on all Borel subsets of the Hilbert cube such that if A and B are isometric, then \(m(A)=m(B)?\) V.8. What is the length of the shortest curve which cannot be enclosed in an open unit equilateral triangle? V.9.3. Let S be a planar set of positive measure. Must there be a point x in S such that the set of distances \{d(x,y): \(y\in S\}\) contains an interval? VII.4. Consider a linear random walk with \(X_ 0=0\) and \(X_ n=\pm n\) with equal probability. Is it true that with probability one the walk goes through every integer infinitely often? Is \(S_ n=X_ 1+...+X_ n\) approximately uniformly distributed over \([- 1/\sqrt{n},1/\sqrt{n}]\)?
scientific article; zbMATH DE number 4045677
scientific article