Here's a cute problem (from Robert M. Young, Excursions in Calculus, p. 244): "What is the average straight line distance between two points on a sphere of radius 1?"
(Answer to follow.)
Note the number of comments with different answers.
"Excursions in Calculus," from what I could see at Google Books, looks like it has many interesting problems.
Beautiful Mathematics is a collection of interesting mathematical explorations published by the MAA. If you find the following questions (many of which are new to me) intriguing then you'll probably enjoy the book.
- Do you know the dimension of Sirpienski's Triangle?
- Have you ever explored squaring maps?
- What is the Riemann Sphere?
- Can you find a formula that associates Fibonacci numbers and Pi?
- Can a square be inscribed in any triangle?
- What are the first three digits of the millionth Fibonacci number?
- Do you know how to construct a regular heptagon using a straight edge, compass, and angle trisecting device?
- Can you prove Lagrange's Theorem, that every positive integer is the sum of four squares?
- How many triangles are there of perimeter n, where n is an integer, the sides are all integer lengths and the triangles are incongruent?
"Beautiful Mathematics" has nearly 100 challenging investigations, most with elegant solutions presented. Topics include words, images, formulas, theorems, proofs, solutions, and unsolved problems.
The book is available from the publisher in PDF format for half of the price of the hardback book.
Deep Math meets great magic tricks. Magical Mathematics is an absolutely remarkable book. I don't say this lightly. Publishers send me plenty of books to review. Some I like more than others. "Magical Mathematics" is a fantastic book for someone who wants to explore the non-trivial math behind some impressive magic tricks.
While I enjoy purely recreational math puzzles that have no practical application I also love it when I discover challenges that are interesting and relevant. "Magical Mathematics" is chock full of fun (and deep) challenges that students (and adults) can sink their teeth into.
This great review by Sami Assaf at Amazon.com gets to the heart of what makes this book stand apart from many other recreational Math books:
The book is packed with fantastic card tricks that will surely dazzle friends and family (with enough practice), but goes beyond this by explaining the beautiful (often deep) mathematics behind the tricks. The book intersperses magic and mathematics in an engaging way that keeps the reader hooked. The book begins with a simple 4 card trick. Well, simple enough to perform; understanding is a different matter. The authors then explain what mathematical concepts (mostly involving combinations and permutations) are at the heart of the trick, and then generalize the principle involved into a truly impressive, more elaborate card trick. After that, it's back to math to see exactly how and why the magic works. Later chapters follow similar patterns, where the reader is drawn in by a beautiful card trick and the beauty is then heightened with a clear explanation of the underlying mathematics. Along the way, the authors give excellent advice on how often to rehearse the tricks before performing as well as entertaining tips to make for a more engaging performance. The mathematics is presented in digestible bites, with excellent examples and illuminating illustrations.
But be warned: this isn't your simple high school math! Many of these tricks employ sophisticated mathematics using Combinatorics, Group Theory, Graph Theory and more. Fortunately, the authors are adept at explaining these complicated concepts in a clear fashion, but the novice reader may have some trouble following some of the proofs. Hopefully, the reader will be so inspired by the beauty of the subject, that she'll see it as motivation to learn more mathematics! In fact, the authors' unapologetic goal with this text is to corrupt youngsters of all ages into pursuing mathematics in much the same way that the authors themselves were seduced by the subject. Here's hoping they succeed with you as they have with me!
The New York Times recently published a review of "Magical Mathematics." Another review is available at The Math Less Traveled. More information about the book is available at the Princeton University Press website.
Google just announced graphing support to its search engine.
More information at the Google Search Blog.
Hat tip to my brother Abe.