Chapter 2

G.3 Write a symbolic statement for each of the following verbal statements. (1) Either Donna does not like Elmer because Elmer is bald, or Donna likes Frank and George because they are handsome twins. (2) Either neither you nor I am honest, or Hank is a liar because Iggy did not have the money.

G.4 Three schools have a track meet and enter one person in each of the events. The number of events is unknown, and so is the scoring system except that the winner of each event scores a certain number of points, second place scores fewer points, and third place scores fewer still. Georgia won with 22, and Alabama and Florida tied with 9 each. Florida won the high jump. Who won the mile run? *

G.5 Consider the following question: "Of all possible collections of states that yield 270 or more electoral votesenough to win a presidential electionwhich collection has the smallest geographical area?" Hint: Let O be the set consisting of the optimal collection of states. Your group should choose a state and prove mathematically that the state is not in O or prove that the state is in O.

References

This problem is found in the following article "Proof by Contradiction and the Electoral College," by Charles Redmond, Michael P. Federici, and Donald M. Platte in The Mathematical Teacher, November 1998.

The U. S. Electoral College Calculator: http://www.bga/jnhtx/ec/ec.html

National Archives and Records Administration Federal Register: http://www.nara.gov/fedreg/ec-hmpage.html

G.6 Five cabbies have been called to pick up five fares at the Hilton Towers. On arrival, they find that their passengers are slightly intoxicated. Each man has a different first and last name, a different profession, and a different destination; in addition, each man's wife has a different first name. Unable to determine who's who and who's going where, the cabbies want to know: Who is the accountant? What is Winston's last name? Who is going to Elm Street? Use only the following facts to answer these questions:

1. Sam is married to Donna.
2. Barbara's husband gets into the third cab.
3. Ulysses is a banker.
4. The last cab goes to Camp St.
5. Alice lives on Denver Street.
6. The teacher gets into the fourth cab.
7. Tom gets into the second cab.
8. Eve is married to the stock broker.
9. .Mr. Brown lives on Denver St.
10. Mr. Camp gets into the cab in front of Connie's husband.
11. Mr. Adams gets into the first cab.
12. Mr. Duncan lives on Bourbon St.
13. The lawyer lives on Anchor St.
14. Mr. Duncan gets into the cab in front of Mr. Evans.
15. The lawyer is three cabs in front of Victor.
16. Mr. Camp is in the cab in front of the teacher.

* From "Ask Marilyn" by Marilyn Vos Savant, Parade Magazine, October 31, 1993.

G.7 Consider the apparatus shown in Figure G.2.



Figure G.2 Reward Game


Note that there are 12 chutes (numbered 1 to 12), and if you drop a ball into the chute it will slide down the tube until it reaches an AND-GATE or an OR-GATE. If two balls reach an AND-GATE, then one ball will pass through, but if only one reaches an AND-gate, it will not pass through. If one or two balls reach an OR-GATE, then one ball will pass through. The object is to obtain a reward by having a ball reach the location called REWARD. What is the fewest number of balls that can be released in order to gain the reward?