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Aha! Solutions (MAA Problem Book Series)

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Mathematicians generally take second place to no other group in appreciating a mind-straining problem that can be solved with the right insight into the structure. In fact, one can argue that that is the primary way that progress is made in mathematics. Although they are split into the levels of elementary, intermediate and advanced, all the problems can be understood if read carefully by someone that has done well in a discrete mathematics course.

The elementary problems will be of special interest to teachers at nearly all levels of high school and college, most can be understood by people with a background in basic algebra with a little set theory and combinatorics included. They could serve as great primers for test and quiz questions as they are problems that appear much more difficult than they actually are.

My favorite in the elementary section is on page 8:

Which is greater sqrt(6) + sqrt(10) or sqrt(5) + sqrt(12)?

And the rules would be that a calculator is not allowed.

The solution is obvious if you simply square both numbers and then compare. Knowing the relative order of positive numbers and their squares gives you the answer immediately. Such problems are also easily altered to appear on exams.

Students in mathematics courses need to learn that there are many routes to an answer or a proof and sometimes, the route is a tedious one through many twists. However, there are times when the solution to a problem comes from somewhere in the mathematical equivalent of "A miracle happens!" This book demonstrates many of those problems in several areas of mathematics.

Published in Journal of Recreational Mathematics, reprinted with permission.