# The Scottish Book

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**PREFACE**

The enclosed collection of mathematical problems has its origin in a notebook which was started in Lwow, in Poland in 1935. If I remember correctly, it was S Banach who suggested keeping track of some of the problems occupying the group of mathematicians there The mathematical life was very intense in Lwow Some of us met practically every day, informally in small groups, at all times of the day to discuss problems of common interest, communicating to each other the latest work and results. Apart from the more official meetings of the local sections of the Mathematical Society (which took place Saturday evenings, almost every week!), there were frequent informal discussions mostly held in one of the coffee houses located near the University building - one of them a coffee house named "Roma" and the other "The Scottish Coffee House". This explains the name of the collection. A large notebook was purchased by Banach and deposited with the headwaiter of the Scottish Coffee House, who, upon demand, would bring it out of some secure hiding place, leave it at the table, and after the guests departed, return it to its secret location.

You can see a picture of the Scottish Café.

Many of the problems date from years before 1935. They were discussed a great deal among the persons whose names are included in the text, and then gradually inscribed into the "book" in ink. Most of the questions proposed were supposed to have had considerable attention devoted to them before an "official" inclusion into the "book" was considered. As the reader will see, this general rule could not guarantee against an occasional question to which the answer was quite simple or even trivial.

In several instances, the problems were solved, right on the spot or within a short time, and the answers were inscribed, perhaps some time after the first formulation of the problem under question.

As most readers will realize, the city of Lwow, and with it the "Scottish Book", was fated to have a very stormy history within a few years of the book's inception A few weeks after the outbreak of World War II, the city was occupied by the Russians. From items at the end of this collection, it will be seen that some Russian mathematicians must have visited the town; they left several problems (and prizes for their solutions). The last date figuring in the book is May 31, 1941. Item Number 193 contains a rather cryptic set of numerical results, signed by Steinhaus, dealing with the distribution of the number of matches in a box! After the start of war between Germany and Russia, the city was occupied by German troops that same summer and the inscriptions ceased.

The fate of the Scottish Book during the remaining years of war is not known to me. According to Steinhaus, this document was brought back to the city of Wroclaw by Banach's son, now a physician in Poland. (Many of the surviving mathematicians from Lwow continue their work in Wroclaw. The tradition of the Scottish Book continues. Since 1945, new problems have been formulated and inscribed and a new volume is in progress.)

A general word of explanation may be here in order: I left Poland late in 1935 but, before the war, visited Lwow every summer in 1936, '37, '38, and '39. The last visit was during the summer preceding the outbreak of World War II, and I remember just a few days before I left Poland, around August 15, the conversation with Mazur on the likelihood of war. It seems that in general people were expecting another crisis like that of Munich in the preceding year, but were not prepared for the imminent world war. Mazur, in a discussion concerning such possibilities, suddenly said to me " A world war may break out. What shall we do with the Scottish Book and our joint unpublished papers? You are leaving for the United States shortly -- and presumably will be safe. In case of a bombardment of the city, I shall put all the manuscripts and the Scottish Book into a case which I shall bury in the ground". We even decided upon a location of this secret hiding place; it was to be near the goal post of a football field outside the city. It is not known to me whether anything of the sort really happened. Apparently, the manuscript of the Scottish Book survived in good enough shape to have a typewritten copy made, which Professor Steinhaus sent to me last year.

The existence of such a collection of problems was mentioned on several occasions, during the last 20 years, to mathematical friends in this country. I have received, since, many requests for copies of this document. It was in answer to such oral and written requests that the present translation was made. This spring in an article, "Can We Grow Geniuses in Science?" which appears in Harper's June, 1957 issue, L. L. Whyte alluded to the existence of the Scottish Book. Apparently, the diffusion of this small mystery became somewhat widespread, and this provided another incentive for this translation.

Before deciding to make such an informal distribution, I consulted my teacher and friend (and senior member of the group of authors of the problems), Professor Steinhaus, about the propriety of circulating this collection. With his agreement, I have translated the original text (the original is mostly in Polish) in order to make it available through this private communication.

Even as an author or co-author of some of the problems, I have felt that the only practical and proper thing to do was to translate them verbatim. No explanations or reformulations of the problems have been made.

Many of the problems have since found their solution, some in the form of published papers. (I know of some of my own problems, solutions to which were published in periodicals, among them, e.g. Problem 17, l. Z. Zahorski, *Fund. Math.* **34** 183-245 and Problem 77(a), R. H. Fox, *Fund. Math.* **34** 278-287.)

The work of following the literature in the several fields with which the problems deal would have been prohibitive for me. The time necessary for supplying the definitions or explanations of terms, all very well understood among mathematicians in Lwow, but perhaps not in current use now, would also be considerable. Some of the authors of the problems are no longer living and since one could not treat uniformly all the material, I have decided to make no changes whatsoever.

Perhaps some of the problems will still present an actual interest to mathematicians. At least the collection gives some picture of the interests of a compact mathematical group, an illustration of the mode of their work and thought; and reflects informal features of life in a very vital mathematical centre. I should be grateful if the recipients of this collection were willing to point out errors, supply information about solution to problems, or indicate developments contained in recent literature in topics connected with the subjects discussed in the problems.

It is with great pleasure that I express thanks to Miss Marie Odell for help in editing the manuscript and to Dr. Milton Wing for looking over the translated manuscript .

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