A few years ago, the author of this book published an index to mathematical problems published between 1980 and 1984, and announced an ambitious program to publish other volumes extending this project both forward and backward in time. I was fortunate to have that volume available during my five-year term as Editor of the Problems and Solutions column of The American Mathematical Monthly. The system for classifying problems by topic, by itself, adds an important level of organization, as does the section on notation, but the Index goes beyond this to allow related problems to be identified, and locate individuals and journals associated with these related problems. The wealth of information about the problems and their means of publication is an enormous service to anyone facing the task of preparing a list of problems.
In his foreword to the 1980-1984 Index, Murray S. Klamkin was critical of the indexing information on problems provided by journals. Although I was in a position to move one journal in the direction that he indicated, there was little change. It thus falls on me to defend the present state of indexing of problems in journals. It is not really an answer to say that I was never asked to develop a better index, since I am sure that anything along these lines that I produced would have been used. The system has some inertia based on the way that various tasks are assigned to meet publication deadlines, although sweeping changes are often made when there is a change of editor. However, the fans of Problem Sections also tend to have strong opinions. Removing the distinction between Elementary and Advanced problems had already generated strong comments: half opposed to the change, and half in favor. While the subject classification of this volume is useful for organizing thousands of problems, it is not clear that such a classification would be useful for fewer than one hundred problems. Indexing by author has the nice feature that it encourages the reader to use the name of the author as the key to locating a distinctive problem or solution. Those whose skill in formulating problems and writing insightful solutions deserve to be closely identified with their work. Additional indexing may well be better confined to indexes of broader scope. The continuation of this project will raise the general level of awareness of this aspect of doing mathematics, and give a better picture of the high value placed on this activity.
The spectrum of problems runs from routine exercises to the great problems capable of inspiring the development of mathematics for a century or more. Those represented here are chosen from a smaller range from contest problems allowing an hour or so to journal problems for which several months of work are needed for an adequate solution. Although this avoids the extremes of the spectrum, there is still room for significant difference in difficulty. Since the reader is expected to be able to solve these problems, it is reasonable to expect that each problem contains the seeds of its solution. Also, full statements of problems are given, so the Index may be enjoyed by someone interested in the subject, as well as (indeed, probably more than) those who need it in their work. A beginner may need guidance in selecting problems suitable to his present level of training, but an experienced mathematician should develop an irresistible urge to pick up pencil and paper after opening the book to a random page in the Subject Index. I would go so far as to suggest that this Index is the ideal retirement gift for a mathematician to allow the fun of the subject to be rediscovered after a career that has reached the point of research on a highly specialized topic and teaching of the same old subjects.
Richard T. Bumby
Professor of Mathematics
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