Jim Totten's problems of the week (Q2842617)

Jim Totten (1947--2008) taught at Saint Mary's University (Halifax, NS) and Cariboo College, later Thompson Rivers University (Kamloops, BC). He was an editor at \textit{Crux Mathematicorum.} For many years he posted weekly problems for the mathematics students in his department. Many of these are collected in this excellent book.NEWLINENEWLINEThe topics are varied; chapter headings are Combinatorial geometry; Functions; Higher dimensional geometry; Identities, inequalities, and expressions; Logic, games, puzzles, and amusements in math; Number theory; Plane geometry; Probability; Triangle mathematics; and Miscellaneous. The level is intermediate: higher than most collections aimed at the general public, but lower than (for instance) \textit{P. Winkler}'s collections [Mathematical puzzles. A connoisseur's collection. Natick, MA: A K Peters (2004; Zbl 1094.00003); Mathematical mind-benders. Wellesley, MA: A K Peters (2007; Zbl 1135.00005)] or books such as \textit{L. C. Larson}'s [Problem-solving through problems. Problem Books in Mathematics. New York etc.: Springer-Verlag. (1983; Zbl 0524.00004)] that aim -- at least in part -- to train competitive problem solvers. Very strong high school students, bright undergraduates, or most professional mathematicians should find the difficulty appropriate.NEWLINENEWLINEAs stated in the introduction, not all the problems are new. I recognized perhaps one in twenty; few readers will suffer too badly from \textit{deja vu}. The price (\$ 68) is perhaps a little high for a recreational book. University and high school libraries should certainly buy a copy; it would also make an excellent prize for undergraduate math contests.NEWLINENEWLINEThe layout is unusual for a problem book, in that the solutions follow the problem immediately, without even a rule or blank line. As a result, it is very easy to accidentally read the first line or two of the solution. Gathering solutions at the end of the chapter or book might have been preferable. I take this to be due to fairly light editing of Totten's original problem files. Another, perhaps related, problem is that many teachable moments are missed: Problem 163 slips past with no mention of dominating sets, and a mention of the ``Dutch book'' concept could well have followed Problem 313. A bibliography, too, would also have been a valuable addition.NEWLINENEWLINEI did notice a few mistakes (to be fair, the excellence of the problems encouraged a rather thorough reading!) Sadly, the first answer to Problem 1 is wrong (\(1\times99\) rather than \(1 \times 100\).) The second diagram for Problem 85 is missing. Problem 214 is a riff on the old ``dividing the horses'' puzzle; the calculation is correct but it should perhaps be mentioned that the ``solution'' of the original puzzle, used in the problem, is no solution. Problem 321 asks for a probability to be determined that can only be estimated. But overall the quality of the problems and writing is very good indeed.NEWLINENEWLINEI would like to end with a little contribution. Problem 357 asks for a proof that if the numbers \(1,2,\dots,36\) are arranged cyclically, some three adjacent values must always add to at least 56. The highest value that can be substituted for 56 is probably to be found in the literature; but for 57, at least, the problem remains an elegant, and still not overly difficult, puzzle.