Let’s begin with some relatively easy problems. The challenges become gradually more difficult as you go through the book. The problems in this chapter can be solved without advanced mathematics. Knowledge of basic arithmetic, algebra, and geometry will be helpful, as well as your own creative thinking. I recommend that you attempt all the problems, even if you already know the answers, because you may discover new and interesting aspects of the solutions. A bonus after each solution discusses a related mathematical topic. Remember, each problem has an aha! solution.

Abby has fifteen cookies and Betty has nine cookies. Carly,...

I hope that you enjoyed the elementary problems. Now let’s try a selection of somewhat more difficult problems. In this chapter, we can expect to use familiar techniques from calculus and other branches of mathematics. As usual, we are looking for illuminating proofs. Aha! solutions are to be found!

At some time between 3:00 and 4:00, the minute hand of a clock passes the hour hand. Exactly what time is this? (Assume that the hands move at uniform rates.)

Let’s solve the problem in a mundane way first (before giving an aha! solution). We reckon time in minutes from the...

An advanced problem may have an aha! solution. This doesn’t necessarily mean that the solution is easy, only that a key step is the product of inspired thinking. You may wish to turn to the Toolkit for some mathematical terms in the statements of these problems. In some cases, the solutions require advanced techniques and concepts.

Given any polygon, join the midpoints of the edges (in order) to produce a new polygon. Notice that the pentagon below yields a “child” (shown with dotted lines) that is non-self-intersecting.

However, the pentagram below yields a child that is self-intersecting.

If we repeat...