ABSTRACT
Preparation for Chapter 4
Prep Problem 4.1. Show that if Left wins moving second on G 1 and G 2, Left can win moving second on G 1 + G 2.
Prep Problem 4.2. Give an example of games G 1 and G 2 such that Left can win moving first on G 1 and on G 2, but cannot win on G 1 + G 2, even if she can choose whether to move first or second on the sum.
Prep Problem 4.3. List the properties (or axioms) you usually associate with the symbols =,+, −, and ≥. One property might be, “If a ≥ b and b ≥ c, then a ≥ c” that is, transitivity of ≥. After spending about 10 to 15 minutes listing as many as you can think of, compare your notes with a classmate.
To the instructor: In an undergraduate class, consider covering material from this chapter interleaved with examples from the next chapter to motivate this more theoretical chapter. In particular, this chapter provides axioms for combinatorial game theory and important theorems that can be derived reasonably directly from the axioms. The next chapter, however, places the axioms in context. (We chose this approach in order to maintain axiomatic continuity.) For a more example-driven development, present the games of hackenbush and cutcake as in Chapter 2 of WW [BCG01] through their section entitled “Comparing Hackenbush Positions,” and assign problems primarily from the next chapter.
The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.
Albert Einstein