My oldest daughter is just about 8 years old. She has told me that she wants to go on a date with a boy in her class (Not going to happen). While humoring her, I asked her why she thinks he would be a good date, and she told me it was because he has a lot of PokÈmon. Now, I know a lot about PokÈmon and how that translates to dateability. In fact, I know that the eort it takes to catch PokÈmon can be expressed as p 2 θ , where p represents the number of PokÈmon someone has caught, and θ represents some private information. θ= 100 for a low quality date (Just a typical PokÈmon trainer) whereas θ= 2311 for a high quality date (a PokÈmon master). Furthermore, a date with my daughter has a payoff associated with it of UD = 275 (She is pretty awesome), but not going on a date with my daughter has a payoff of UN = 50. My daughter only likes to date those who are of high quality and would prefer to reject those of either low or unknown quality.

(a) How many PokÈmon would a high quality date be willing to catch to earn a date with my daughter? (This won't be an integer)

(b) How many PokÈmon would a low quality date be willing to catch to earn a date with my daughter?

(c) How many PokÈmon must a high quality date catch in order to effectively signal their type?