**Solutions to Exercises**

**1.** *Answer:* 2. There is strong evidence that human babies, chimpanzees, and even rats have an understanding of this problem. See S. Dehaene, *The Number Sense: How the Mind Creates Mathematics*, Oxford, New York, 1997, pp. 23–27, 52–56.

**2.** Here is one possible way to arrive at the answers. There may be other equally simple (or even simpler) ways to solve these problems.

**3. a.** Given any *n*-subset *S* of [*x* + *n* + 1], there is a largest *k* for which #(*S* ∩[*x* + *k*]) = *k*. Given *k*, we can choose *S* to consist of any *k*-element subset in ways, together with {*x* + *k* + 2,*x* + *k* + 3, …,*x* + *n* + 1}.

**b.** *First proof.* Choose a subset of ...

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