Creative Thinking

Answer these questions thoroughly and clearly, as if you were explaining them to another student. Diagrams are optional. Each question is worth 8 points except #6 and #8 which are worth 10 points each.

1. The Moat
   You are trying to enter a castle surrounded by a moat that is 25 feet wide. There is no draw bridge and the moat is filled with piranha. The only tools available to you are two 24 foot wooden planks. How do you reach the castle?

2. Hands of a Clock
   How many times within a 24 hour period will the hour hand and minute hand of a clock point in the exact same direction. Try it from 6:00 a.m. Monday to 6:00 a.m. Tuesday.

3. Exactly Six Pints
   You have two pitchers. One holds 4 pints and the other holds 9 pints. How do you bring back exactly 6 pints of water from a river if you have no other tools and you are not allowed to estimate the amount of water in a pitcher. (Note: Filling a pitcher halfway full is estimating.)

4. Cutting Pizza
   You want to cut a pizza into as many pieces as you can. What is the greatest number of pieces you could create given only:
   1. 2 cuts
   2. 3 cuts
   3. 4 cuts
   4. 10 cuts

5. Crossing the Bridge
   Four people need to cross a bridge at night. The bridge is only strong enough to hold at most two people at once. Because it is night a flashlight must be used for all crossings. It takes person A ten minutes to cross, person B five minutes, person C two minutes, and person D one minute. If two cross at the same time they must walk at the slower man's pace. How can you get everyone across in 17 minutes?

6. The Greatest Garden
   A farmer uses 1000 feet of fence to enclose and subdivide a rectangular garden. What dimensions will produce the greatest enclosed area?

7. The Chain Bracelet
   A woman flies to a tropical island for a week long vacation. However, when she arrives at the hotel she suddenly realizes that she has lost all her cash and credit cards and has no way to pay for her stay. She tells this to the hotel manager, explaining that she could have the money wired there by a friend in one week. She asks him if it would be okay to pay at the end of the seven days rather than at the beginning. The hotel manager agrees to this as long as he receives some form of collateral. Noticing a seven-link silver chain bracelet she is wearing, he explains that this will cover the cost and suggests that he keep it until the money comes in. The woman agrees but argues that she should only have to pay what she owes on a particular day. The first day she would pay one link, and on each day following he would receive an additional link. Sadly this means that the bracelet must be cut so that the links may be separated. Wanting to do as little damage to the bracelet as possible, what is the fewest number of cuts she could make to be able to pay for her stay?

8. Two Candles
   There are two candles. Both are the same height, but one is twice as wide as the other. The thinner candle will burn all the way down in 3 hours. The thicker one requires 6 hours. If both candles are lit at the same time, how long will it take for the thicker candle to be four times the height of the thinner candle?

9. Buried Cable
   There is a long, straight cable buried under a unit square field. You must dig one or more ditches to locate the buried cable. You do not know the direction of the cable--north/south, east/west, diagonal, etc. It is possible that the cable just barely crosses under one corner of the square field, in which case you would have to dig near that corner to locate it. Where should you dig to guarantee finding the cable and to minimize digging?

10. Fair Voting
    A certain company makes all executive decisions by asking its stockholders to take a vote. Each stockholder’s vote is weighted proportional to the amount of stock they own. For example, someone who owned 3% of the stock would have 3 votes. There would be 100 total votes. A majority is needed to win. The stock is divided as follows:

    51% is owned by person A
    25% is owned by person B
    12% is owned by person C
    6% is owned by person D
    3% is owned by person E
    2% is owned by person F
    1% is owned by person G

    Is this a fair voting system? Why or why not? If not, devise a voting system that is more fair than this one if it can be done.

11. The Prisoner's Dilemma
    You and an associate of yours named Stu are caught by the police near the scene of a crime. You are both taken to headquarters and are separately interrogated by the police. Both of you are urged to confess the other’s guilt. You are told the following:
    
    

    	Stu confesses your guilt	Stu does not confess your guilt
    You confess Stu's guilt	You both get 10 years in prison	You go free Stu gets 20
    You do not confess Stu's guilt	You get 20 years Stu goes free	You both get 1 year in prison

    Given that your only goal is your own freedom, what should you do? Explain.
    How would your action be different if you were able to communicate with Stu?