# Warm-Up 10, Due Monday, October 7, 11:59 pm.

Warm-up questions usually either address material we handled during last class, or material you were to read in preparation for class, or you are supposed to answer questions based on your common sense alone. Please answer each question in three or more sentences. It is OK to answer 'I don't know' - but try to tell me why you are confused! No late Warm Ups are accepted for any reason, and only those submitted electronically through this web page (or by email, if the web page has technical problems) are considered.

## Question 1

Choose one of the three values p=0.15, p=0.5, or p=0.7. Please adjust the two values of n=6 and your p in the two textboxes at the bottom of the applet. Then play 10 rounds of RANDOM NIM(6,p) with 6 stones and the probability you have chosen above. This probability is the probability that a stone is taken between the rounds of you and the computer taking some stones. After this, you may have a feel for the game. Then play 20 rounds. Tell me the result of these 20 rounds. How much did you win, how much did you lose? Explain how you played. Did you start taking one or two stones?

If the applet doesn't work, just tell me.

## Question 2

Five pirates have to decide decide how to distribute 100 gold coins they have found. They do it according to the following rules. In each round they sit in a circle and rotate a bottle of rum. The one to which the bottle pointgs has to propose a distribuation of the coins. This proposal is voted on, with the proposer voting too, and even deciding in case of a tie. That implies that in case of only two pirates, the proposal will always win, and in case of four pirates, the proposer needs only one of the others voting for the proposal. If the propsal wins, everything is settled. If not, the proposer is thrown into the water and dies, and the next round starts. Assume that the pirates value their life worth 200 gold coins, and prefer throwing somebody overboard if everything else is equal (are playing the "hostile" variant).

What would you propose if the bottle points on you and there are 2, 3, 4, or 5 pirates left?

## Question 3

We play the variant of the Pirates Game discussed above, but this time the pirates are really fierce, and consider their own life only to be worth 10 gold coins.

What would you propose if the bottle points on you and there are 2, 3, 4, or 5 pirates left?