**Math 102 Final**

**June, 2001**

1. You are playing a two-person game with a heap of 21 beans. The players alternate removing 1,2,3 or 4 (!) beans from the heap. The player to remove the last bean(s) wins. Should you go first or second? Describe a winning strategy.

2. You are on a game show where 1 million dollars is concealed behind one of 4 doors. You guess door #1 and the game show host opens doors #2 and #3 to show you the money is not there. Then she offers you the choice of switching your guess to door #4. Should you switch? Specifically explain your chances of winning under the best strategy.

3. A dating service is working with 8 divorced couples. They wish to hold a series of parties in which each female gets to meet all the other males. Of course, no female will attend a party with her ex-husband. What is the fewest number of parties that need to be held?

4. You need to cook 6 grilled cheese sandwiches on a grill that has room for exactly two sandwiches. To grill the sandwiches takes one minute on each side, 1 second to flip, 2 seconds to put on or take off a sandwich. What is the least amount of time in which you can grill all 6 sandwiches?

5. Explain the strategies represented by the letters PSSSP. Under each strategy, briefly list and explain the substrategies for solving interpersonal problems that we discussed in class.

6. (a) Solve the cryptarithmetic problem EGG + EGG = PAGE.

2 1

(b) There are
two cards with a number on each side lying on the table.

Which cards must you turn over to verify the statement: “All cards with an even number on one side have an odd number on the other side”?

7.
On an 8 by 8 checkerboard, players alternate placing markers
in the squares. The player who goes first places a marker in the lower left
corner square. The next player can place a marker either in the square one *above* or in the square one to the *right* of the marker just placed on the
previous move. The player to place a marker in the upper right corner wins.
What is a winning strategy? Explain any analogies to previous problems you have
solved.

8. You are at a desert oasis with 800 liters of gasoline. An outpost 1000 kilometers away desperately needs gasoline. Your small jeep can carry at most 200 liters of gas and it burns one liter for every 5 kilometers it travels. How many liters of gasoline can you deliver to the outpost? (At any time you may drop off gas in cans and return later to pick it up.)