Ogilvy C.S. Tomorrow's math: Unsolved Problems for the Amateur

2nd. edition. — Oxford University Press, 1972. — vi, 198 p. — ISBN: 0-195015088, 978-0195015089.When there are many people who don't need to expect something more than the benefits to take, we will suggest you to have willing to reach all benefits. Be sure and surely do to take this tomorrows math unsolved problems for the amateur that gives the best reasons to read. When you really need to get the reason why, this tomorrows math unsolved problems for the amateur book will probably make you feel curious.
This is the second edition of a book which first came out in 1962. It is a collection of over 150 unsolved mathematical problems, selected and presented so that amateurs can understand them.
Inevitably most of the problems are mathematically trivial, and the author includes a word of apology for his title: “Not by any stretch of the imagination could this book be considered a summary of those topics that will be the major concern of tomorrow’s mathematicians.” The problems are drawn from several branches of mathematics, and vary from famous problems such as Fermat’s Last Theorem and the Four Colour Problem to curiosities such as “prime primes” (317 is an example of a “prime prime”, because 317, 31, and 3 are primes, successively dropping digits from the right — provided we are in base 10).
The author’s occasional simplifications are liable to be misleading, for example in his brief discussion of Godel on pp. 20, 21, and even more so in his statement on p. 7 of results due to Banach, Tarski and Robinson: “It is possible to cut a solid sphere into a finite number of pieces and reassemble them by rigid motions (no distortion) to form two solid spheres (no holes), each of the same size as the original one... there need be no more than five pieces!.. There cannot be any completely general definition which will preserve volume under rigid motion.” Here one wonders whether the startling effect is being deliberately sought by the use of the words “cut” and “pieces” to describe a subdivision into five point sets, four of which are not measurable ; the result is startling enough when properly stated. Another minor mathematical criticism is the number of problems in which a formula is required as the solution to the problem when, as the author once or twice mentions, such a formula may well not exist.
Extensive notes at the end of the book give references and further discussion, but unfortunately there is no forward referencing to these notes in the main text. There are several misprints, none very serious, and the arrangement of the contents on the outer cover does not correspond exactly with that in the book — presumably it refers to the first edition. Other small errors give the impression of somewhat hasty composition or revision.Preface to the second edition
The meaning of an unsolved problem
Applied problems
Problems concerning games
Geometrical problems
Arithmetical problems
Topological problems
Probability and combinatorial problems
A glimpse of some problems of analysis
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