Homework 7, due Monday, January 31, 2011

[1] Reading from our book: Read "Number Personalities" and "Hardy and Ramanujan", pp. 43-44.

[2] Turn in the following: Let's say a certain car dealership has 100 cars of the same model on its lot.
• (a) Suppose this model comes in 6 colors, but in all other respects all cars of this model are identical.
• (i) If someone comes in and wants to buy two of these cars with the same color but doesn't care what that color is, explain how the salesperson knows on the spot without checking that the dealership has two such cars to sell.
• (ii) Suppose the customer will buy as many of these cars as they have on the spot, as long as they all have the same color, whatever that color may be. How many can the salesperson be sure that they have in one color, without even looking? Explain your answer.
• (b) Suppose instead in addition to there being a choice of any of 6 colors, that there are two options (manual or automatic, and moonroof or no moonroof).
• (i) Can the salesperson still be sure that they have at least two identical cars on the lot? Explain why or why not.
• (ii) What's the largest number of identical cars that the salesperson can be sure they have? Explain your answer.