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!

Algebra

Passing Time

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.)

Solution

Let's solve the problem in a mundane way first (before giving an aha! solution).We reckon time in minutes from the top of the hour (12 on the clock). Suppose that the minute hand is at t minutes and the hour hand is, correspondingly, at 15+t/12 minutes. (The term t/12 is due to the fact that in 60 minutes the hour hand advances 5 minutes.) When the two hands coincide, we have t = 15+t/12, and hence t = 16+4/11. Therefore the solution is 3:16+4/11 minutes.

In order to delve deeper into this problem, let's answer the same question for the time between 2:00 and 3:00 when the minute hand and hour hand coincide. Using the same reasoning as before, we obtain t = 10+t/12, and hence t = 10+10/11, yielding the solution 2:10+10/11 minutes. We see fractions with 11 in the denominator in both cases. Reflecting on this, we realize that in the course of twelve hours the minute hand and hour hand coincide eleven times.