Jewish Problems

This is a special collection of problems that were given to select applicants during oral entrance exams to the math department of Moscow State University. These problems were designed to prevent Jewish people and other undesirables from getting a passing grade. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is difficult to find. Using problems with a simple solution protected the administration from extra complaints and appeals. This collection therefore has mathematical as well as historical value. 1 A personal story of Tanya Khovanova In the summer of 1975, while I was in a Soviet math camp preparing to compete in the International Math Olympiad on behalf of the Soviet Union, my fellow team members and I were approached for help by Valera Senderov, a math teacher in one of Moscow’s best special math schools. The Mathematics Department of Moscow State University, the most prestigious mathematics school in Russia, was at that time actively trying to keep Jewish students (and other “undesirables”) from enrolling in the department. One of the methods they used for doing this was to give the unwanted students a different set of problems on their oral exam. I was told that these problems were carefully designed to have elementary solutions (so that the Department could avoid scandals) that were nearly impossible to find. Any student who failed to answer could easily be rejected, so this system was an effective method of controlling admissions. These kinds of math problems were informally referred to as “Jewish” problems or “coffins”. “Coffins” is the literal translation from Russian; they have also been called “killer” problems in English. 1 ar X iv :1 11 0. 15 56 v2 [ m at h. H O ] 1 5 O ct 2 01 1 These problems and their solutions were, of course, kept secret, but Valera Senderov and his friends had managed to collect a list. In 1975, they approached us to solve these problems, so that they could train the Jewish and other students in these mathematical ideas. Our team of the best eight Soviet students, during the month we had the problems, solved only half of them. True, that we had other priorities, but this fact speaks to the difficulty of these problems. Being young and impressionable, I was shaken by this whole situation. I had had no idea that such blatant discrimination had been going on. In addition to trying to solve them at the time, I kept these problems as my most valuable possession—I still have that teal notebook. Later, I emigrated to the United States. When I started my own web page, one of the first things I did was to post some of the problems. People sent me more problems, and solutions to the ones I had. It turned out that not all of the coffins even had elementary solutions: some were intentionally ambiguous questions, some were just plain hard, some had impossible premises. This article is a selection from my collection; we picked out some choice problems that do contain interesting tricks or ideas. Tanya Khovanova 2 Introduction Discrimination against Jewish people at the entrance exams to the most prestigious universities in the USSR is a documented fact, see A. Shen  and A. Vershik . Alexander Shen in his article published 25 problems that were given to Jewish applicants. Later Ilan Vardi wrote solutions to all those problems, see I. Vardi’s homepage . These and other articles were published later in a book, “You Failed Your Math Test,  Adventures and Misadventures of Young Mathematicians, Or Test Your Skills in Almost Recreational Mathematics.” . In the present collection we have only one problem that overlaps with the problems that Vardi solved, and we give an easier and more surprising solution. Now, after thirty years, these problems seem easier. Mostly, this is because the ideas of how to solve these problems have spread and are now a part of the standard set of ideas. Thirty years ago these problems were harder to solve and, in addition, the students were given these problems one after another until they failed one of them, at which point they were given a failing mark. To give readers an opportunity to try and solve these problems we separated problems, key ideas, and solutions. Section 3 contains 21 problems. In the subsequent Section 4 we give hints or main ideas for the solutions. Finally, Section 5 contains solutions.