This chapter might be called "counting around a circle" as opposed to "counting on a line." Ordinary counting can be viewed as moving a marker one unit along the number line for each object counted. We can do the same thing on a circle instead, and that is precisely the principle underlying the clock, where the hour marker moves one unit around the circle for each hour that passes. Hence the phrase "clock arithmetic." The material in this chapter relates to some basic ideas of abstract mathematics: algebraic structures, generalization, conjecture and proof, the pigeonhole principle, etc. However, these ideas are all expressed in terms of simple numerical computations.

10.1 The twelve-hour clock Clock arithmetic

10.2 Arithmetic of even and odd; casting out nines

10.3 Zero divisors

10.4 Pigeonholes and inverses

10.5 The perfect shuffle

10.6 Fermat’s little theorem