No calculus or anything beyond is required to read this book. A bright high school student should find it quite accessible. It would serve as good training for math contests. Many of the problems are difficult and should be tackled by someone with strong mathematical skills.
This book, originally written in 1967 and later corrected and republished in 1985, is a collection of 270 mathematical problems and puzzles from the fields of arithmetic, algebra, plane and solid geometry, trigonometry, number theory, and recreational mathematics, including dissections, cryptarithms, and magic squares. The problems offer a lot of variety, both in terms of subject matter and difficulty level.
The author was editor of the Problems and Questions department of Mathematics Magazine, where he introduced the subdepartment entitled "Quickies." He presumably compiled some problems from this source along with other published sources and problems of his own invention. Sources are given for all solutions that are not his own.
The point of the problems is that while most can be solved using long, laborious, or advanced techniques, there are short, elegant, and elementary solutions possible. Many of the solutions contain an element of surprise. The author challenges the reader to "devise a neater, quicker, more elegant solution than the one published in the solution section." An additional challenge is provided by not classifying the problems as easy or hard. They are presented in a random order with no indication as to the difficulty level.
A few problems are well-known theorems: "Prove that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides;" "Show that there is an infinitude of prime numbers;" "In any cyclic quadrilateral, show that the product of the diagonals equals the sum of the products of the opposite sides." Some of the problems are versions of well-known puzzles: "There are n players in an elimination-type singles tennis tournament. How many matches must be played (or defaulted) to determine the winner?"; "Three segments, 3, 4, and 5 inches long, one from each vertex of an equilateral triangle, meet at an interior point P. How long is the side of the triangle?" But most problems are original or not well-known (at least to this reviewer).
This book made for highly entertaining bed-time reading. I often fell asleep thinking about the problems. I also solved a couple while driving to work in the morning. But mostly, I was impatient and just immediately read the solution, not having the time to invest in trying to solve them all. The solution was almost always surprisingly short and elegant, so my entertainment was hardly diminished by skipping right to the solution.
Often, I came across a problem that I had previously solved, or could solve, using calculus, but the solution given by the author would cleverly avoid calculus. So, this was quite pleasant. Also, there were some theorems for which I knew a proof, but the book would give me a new and often better proof. For example, in any triangle, an angle bisector divides the opposite side into two segments proportional to the the other two sides of the triangle. I knew a standard proof of this involving drawing auxiliary lines and using similar triangles, and I also came across a proof using the law of sines, but the solution given offered a much more natural and simpler proof that now makes me wonder why I never knew it before.
I'll describe a few of the more entertaining problems (without giving away solutions). One problems asks for a proof that no regular polygon of more than four sides can be inscribed in a non-circular ellipse. Another problem posits an even number of points constituting a bounded set in a plane, and asks for a line that passes through none of them such that exactly half of the points are on one side of the line. Yet another problem asks for the curve of minimum length which divides an equilateral triangular region into two parts of equal area. There are many more entertaining questions.
The book did have some unpleasant, boring, and silly questions as well, such as the major theorems mentioned above. There were also some mental arithmetic questions such as "Square 85 mentally" that were probably a bad idea to include in a book like this. A couple of themes were a little repetitive too, such as the author's penchant for questions involving different number bases. But overall, these bad questions didn't bother me because they were easily skipped. There was always another good question one could turn too.