“There is an immense amount of knowledge lying scattered at the present day, and almost useless from the difficulty of finding it when wanted.” — Professor J. D. Everett
The above quotation comes from George S. Carr’s book, “Synopsis of Elementary Results in Pure Mathematics” which was published in 1886. The statement is almost as true today as it was over a hundred years ago. We have come a long way in the last few years. We now have computers and databases that allow us to search for mathematical information from a large variety of sources. The MathSci database, organized by the American Mathematical Society, allows one to quickly search many years of Mathematical Reviews looking for desired information. There are several other journals that abstract mathematical information such as Zentralblatt für Mathematik and the Compumath Citation Index. Yet despite all these abstracting journals, there is still a wealth of mathematical information buried within the abstracted articles that is still unavailable to all but a few tireless researchers.
One of the areas that has been least cataloged are the many mathematical problem columns that appear in various journals from around the world. Beginning on page 439 in this book, we list over 200 journals that have at one time or another had problem columns. There is a wealth of mathematical information squirreled away in these problem columns. There is no easy way to access this material since it is not currently being reviewed, cataloged, or abstracted by anyone. Frequently, I will come across a problem during research or recreation and vaguely remember that this problem (or something similar) was already solved in some journal a few years ago – but then I cannot remember which journal or even the date. There is no way to retrieve this information.
Even the editors and ardent problem solvers who follow these problem columns regularly have a hard time remembering problems that have been solved less than 10 years ago. Often a problem will get published in a journal, and then when the time comes to print the solution, the editor will remark that an observant reader reported that this problem had been proposed and solved in that same journal years earlier. Most solvers either forgot or had no way of knowing in the first place that the problem was not new. What chance would they have had to have located the problem had it been published 50 years ago and in a more obscure journal? There are many beautiful results hidden in the problem column literature. There must be a large amount of effort duplicated by modern researchers, problem solvers, and amateur mathematicians when they are forced to reprove some result that does in fact appear in the literature, but is not readily accessible.
I am an avid problem solver and proposer and get very frustrated when I am trying to solve some problem and have a dim memory of a related result that I need, but cannot remember the reference. I am also amused when someone publishes what they think is an original result, only to be later informed that the result was previously published as a problem in The American Mathematical Monthly in the 1920’s. Léo Sauvé, former editor of Crux Mathematicorum, an international problems journal, once quipped that it seemed like all problems “had once been published in the Monthly”. There are also tens of thousands of problems published in foreign-language journals such as Mathesis and Gazeta Matematica that are virtually unknown to the English-speaking world. Going back in time, we find renowned problem journals such as Mathematical Questions and Solutions from the Educational Times that have also had tens of thousands of problems and solutions published, yet they are little known today.
This index is my attempt to remedy this deficiency. I am starting with a 5-year index, because a larger span of years would have produced a book that would be too large to be manageable. This index covers the mathematical material that can be found in various problem columns from 28 journals from around the world that were publishing problems during the years 1980–1984. It also includes problems from national and international mathematical competitions held during those years. Unfortunately, it is not easy to summarize this mass of data. Unlike Mathematical Reviews, that can publish a short abstract of a long piece of mathematical research; when a problem is only a few lines long to begin with, there is no easy way to summarize this information. I gave a lot of thought about how to produce a meaningful summary and decided that it just wasn’t useful to abstract a 4-line problem into just a few phrases such as “problem about summing tangents of half-angles”. In most cases, such a concise summary was just not good enough to let me really home in on the problem I was looking for. In many cases, such a summary was almost as long as the problem itself. I decided that the only practical way of indexing these problems was to reproduce the text of the problem itself. These problems could then be classified according to mathematical content and then sorted and printed by topic. Thus, to find a particular result about summing tangents of certain angles, you would scan down the subject classification list to “trigonometry”, then scan further for “finite sums”, and then turn to that page of the index where the problems about finite trigonometric sums were printed. Scanning around in this portion of the index would then readily let you find the problem you were looking for. You would also easily find other related problems at the same time that might aid you in your research.
The current index is a small start at solving the complete problem of indexing the world’s math problems. It indexes about 5000 problems published during the years 1980–1984. These problems are all in English. I had intended to publish English translations of problems from other journals that published during those years, such as Elemente der Mathematik, Gaceta Matemática, Gazeta Matematica, Kömal, Kvant, Nieuw Tijdschrift voor Wiskunde, Normat, Praxis der Mathematik, Revue de Mathématiques Spéciales and Matematika v Shkole, but the amount of material became too large to fit in a single volume. I will therefore cover these journals (and others) in another volume. If these indexes are well-received by the mathematical community, I will then continue to do additional 5-year indexes such as 1985–1989 and 1975–1979 and then work backwards 5 years at a time. At some point, this data can be published as a computer database, to facilitate searching for specified topics.
In addition to the problems sorted by topic, I have also created an author index for those times when you remember a result published by a particular author and need help locating it. There is also a title index and a keyword index to further help you locate material when you know a word or phrase that might occur within the problem. Readers will also find the citation index extremely useful. It lists articles published during the years 1980–1984 that reference problems from a problem column or contest.
This index is not a conventional problem book, although it is certainly a rich source of problems. No solutions are published. This index is not meant to replace the many journals and problem columns that contain the original solutions and results. Once researchers have located the problem that they are interested in, the problem chronology allows them to find the name of the journal, volume and page number where the problem was published and lets them go back to this journal to look up the solution. In many cases, the problem printed in this index is only a summary of the complete problem or result and the original reference should be consulted. There the researcher will also find supplementary information and possibly even references to related results.
I hope that the mathematical community will find this index a significant contribution to the mathematical indexing literature. Suggestions are welcome for future indexes and we are certainly interested in hearing about errors in this index, since the data can be corrected in our computer database and corrections published as errata in future volumes. MathPro Press would also like to hear about journals that we have missed. We would also enjoy hearing from readers who would like to help out in future indexing ventures, either by locating or supplying obscure journals that should be indexed or as proofreaders, typists, translators, citation scanners, or problem classifiers.
Even if you’re not a problemist, I’m sure you will enjoy, as much as I do, just browsing through the many pages of interesting results contained herein.
Stanley Rabinowitz
MathPro Press
Chelmsford, Massachusetts
Email address: stan@MathProPress.com