This is a collection of problems and solutions of miscellaneous mathematical problems at an advanced undergraduate level. They are selected from the notes of Jim Totten (1947-2008) after he passed away unexpectedly. Jim Totten was problem editor and later editor in chief of Crux Mathematicorum of the Canadian Mathematical Society.
Jim Totten was problem editor and later editor in chief of the journal Crux Mathematicorum of the Canadian Mathematical Society. In 1986 he collected 80 of his problems under the title Problems of the Week as volume 7 of the ATOM (A Taste of Mathematics) series published by the CMS. The origin of the title is that when he started teaching in 1976, he posted a weekly problem to challenge his students. The response to these was so positive that it tempted him to continue the idea for about 30 more years.
The present book contains 406 problems and solutions that were collected from Totten's notes that he left after his unexpected death in 2008. Although, to solve the problems, the required mathematics are at an undergraduate level, covering many different topics such as logic, geometry, functions, number theory, statistics, etc., the problems are often quite challenging. Even for professional mathematicians they are not at all trivial. They are often formulated as brain teasing puzzles with an underlying recreational flavour. Whatever the formulation is, the solution always requires some sound mathematical reasoning. The witty solutions are included and often require more than just the application of standard class room recipes. Not that they are terribly complicated, once you know the trick, but it may take a pencil and possibly several pages of trial an error to arrive at the key that will set you on the rails to find the solution. Since Fermat's margin note, we know that a problem with a simple formulation can take a highly advanced body of mathematics to solve a seemingly simple problem. Not in the case of these problems. All what is needed stays well within the package that an undergraduate student should be able to deal with. Just a clear and agile mind that is sensitive for this type of puzzles suffices.
The kind of problems can best be illustrated by giving two examples that I selected just because they are short to formulate. Here is one from number theory: "How old is the captain, how many children has he, and how long is his boat, given the product 32118 of the three desired integers? The length of his boat is given in feet (several feet), the captain has both sons and daughters, he has more years than children, but he is not yet one hundred years old." And here is one from plane geometry: "Three circles or radius r each pass through the centers of the other two. What is the area of their common intersection?".
Solving all of the problems will provide many hours and days of puzzling pass-time. There is however the temptation to read the answer prematurely because the proposed solution immediately follows the problem formulation, enticing the reader to read it before he or she even tried to work it out him- or herself. If the purpose is that the reader should indeed find the solutions, then it might have been a better idea to give all the problems in the first part and all the solutions in a second part. However, if the reader is looking for problems to give to others and at the same time estimate the level of difficulty, then the present format is of course the better one. It is technically also the simpler solution because many of the geometric problems require drawings, which should then be repeated for problem and answer to avoid an annoying flipping back and forth to match the text with the figure.
There are of course many other popular puzzler books on the market that also have a lot of brain teasers and logical recreations, but these are usually requiring less mathematics and less computation. Martin Gardner's mathematical puzzles come close, but there the emphasis is often on the recreational aspect, rather than on the mathematical side. In Totten's problems the scales tip much more to the mathematical side.