301 Initial Project


Complete eight of these questions and compose two of  your own for the problem solving project.  Write complete solutions including an analysis of  your problem solving strategy.  Please diversely select your questions to complete.  If your questions are too similar, it may negatively affect your evaluation.  Each question will be scored out of nine points with an additional ten points evaluation for the difficulty and diversity of your selection of questions. 

1.  I had a package of Saran Wrap with the following information:  250 sq. ft. (256 ft x 11 3/4 in), "Bonus 25% Free" and "Improved 30% thicker than before".  How does the current volume compare to the original volume?  I was curious, so I measured the diameter of the paper roll on which the product is wrapped to be 1 1/2 in.  What can you say about the comparative thickness of the roll over the old roll?  Discuss anything you can. 

2.  Estimate how much water is used on campus during a week that students are present.

3.  I regularly receive an offer in the mail which says "Now we can save you up to $327.96 or more a year on auto insurance."  Discuss.  Discuss "I could care less."  Discuss "That's not fair."    Discuss "If you pick a guinea pig up by its tail, then its eyes will pop out." 

4.  It's 3a and starts snowing.  The snow picks up and by 5a the entire Wegmans complex parking lot is covered with 3" of snow.  At this time the expert snow relocation team is called out.  They proceed to move and mold the snow until it is a perfect conical pile.  What is the height of this pile of snow?

5.  People on the island of St. Martin's have either blue or brown eyes.  The peculiar law of St. Martin's is that anyone who discovers that he or she has blue eyes must leave at mightnight the day of discovery, never to return.  No one ever leaves for any other reason.  The catch is no islander is allowed to discuss, or communciate in any way at all, anything about anyone's eye colour.  Furthermore, there are no mirrors, camera, or other devices by which one could directly find out the colour of one's own eyes.
    They all know each other and they are all excellent at reasoning.  After generations of isolation, one day a mainland visitor appears among them.  she says to an assemblage of all the islanders, "I see blue eyes here."  She speaks the truth, and everyone believes her.  What happens?   Explain and justify your answer.  

6.  What is your density?  Is this answer reasonable?

7.  How many new faculty members would be needed, and what would the total cost be to the college, in order to make sure that all classes have twenty or fewer students?  How much would taxes have to be raised in NYS to provide public school teachers with twice their present salaries?

8.  Carmilla Snobnosey lifted the delicate Spode tea pot and poured exactly three ounces of the aromatic brew into the flowered, shell china teacup.  She placed the cream pitcher, also containing exactly three ounces, on the Revere silver tray and carried the offering to Podmarsh Hogslopper.
   "Would you like some tea and cream, Mr. Hogslopper?" she asked.
   "Yup.  Thanks.  Ow doggie, sure looks hot.  I'd better cool it down with this here milk," he responded and carefully poured exactly one ounce of cream into his steaming tea and stirred.  "That oughta do it," he said when the steam stopped rising from the tea.  "Here, I'll just give you back that there cream."  Whereupon he carefully spooned exactly one ounce from his teacup back into the creamer.  Podmarsh blushed as he looked at a tea leaf or two floating in the cream and realised his faux pas.  Caught at an awkward pass, he diced to smooth things over with an intriguing puzzle. 
   "Ya know, Mrs. Snobnosey, I wonder if the tea is more diluted than the cream, or whether the cream is more diluted than the tea?" 
   Resolve the dilution problem. 

9.  For his 13th birthday, Adam was allowed to travel down to Sarah's Sporting Goods store to purchase a brand new fishing pole.  With great excitement and anticipation, Adam boarded the bus on his own and arrived at Sarah's store.  Although the collection of fishing poles was tremendous, there was only one pole for Adam and he bought it:  a 5-foot, one-piece fiberglass "Trout Troller 570" fishing pole.
   When Adam's return bus arrived, the driver reported that Adam could not board the bus with the fishing pole.  Objects longer than 4 feet were not allowed on the bus.  Adam remained at the bus stop holding his beautiful 5-foot Trout Troller.  Sarah, who had observed the whole ordeal, rushed out and said, "We'll get your fishing pole on the bus!"  Sure enough, when the same bus and the same driver returned, Adam boarded the bus with his fishing pole, and the driver welcomed him aboard with a smile.  How was Sarah able to have Adam board the bus with his 5-foot fishing pole without breaking or bending the bus-line rules or the pole?

10.  (This question comes from a former 301 student).  A fly that cannot fly is in the bottom corner of a shoe box whose dimensions are 6" x 4" x 12".  What is the shortest distance that the fly must walk to reach the upper corner on the opposite side?

11.  You have two hourglasses:  a 4-minute glass and a 7-minute glass.  You want to measure 9 minutes.  How do you do it?

12.  Arrange 9 squares with sides measuring 1, 4, 7, 8, 9, 10, 14, 15 and 18 to form a single rectangle. 

13.  John was trying to take a short cut through a very narrow tunnel when he heard the whistle of an approaching train behind him.  Having reached three-eighths of the length of the tunnel, he could have turned back and, running at ten miles/hour, cleared the entrance of the tunnel just as the train entered.  Alternatively, if he kept running forward, the train
would have reached him the moment he got to the tunnel exit.  At what speed was the train moving?

14.  Estimate the volume of salt currently at the American Rock Salt plant south of Geneseo. 

15.  Compute anything you can about King Kong in the 2005 film.  Measurements of length (height, arms &c), volume, weight, and area all seem reasonable.

16.  You have twelve identical-looking stones and three balance scales.  Each scale is clearly labeled "One Use Only."  You are told "A diamond is embedded in one of the stones.  Eleven of the stones weigh the same, but the stone containing the jewel weighs either slightly more or slightly less than the others.  I am not telling you which - you must find the right stone and tell me whether it is heavier or lighter."

17.  Ms. Townsend, the math teacher, knew she had her hands full when it came to parent conference day.  Three sets of parents--the Abercrombies, Balderdashes, and Cockamamies--were all unusually competitive, and each had a child in Ms. Townsend's algebra class.
    So when the Abercrombies came around, Ms. Townsend assured them that their child usually performed better on test than the Balderdashes' child.  She then turned around and told the Balderdashes that their child usually performed better on tests than the Cockamamies' child.  And she told the Cocamamies that their child usually performed better on tests than the Abercrombies' child.  
    How is this possible?  What does this have to do with challenges arising in three-candidate elections?  

18.  Consider a closed-in pen containing 16 hungry lions and a single solitary sheep.  If you are fearing for the sheeps' life, your fears are well placed:  all things being equal, any one of the lions would gladly eat the sheep.
    Ah, but there is a snag.  If any one of the lions devours the hapless sheep, that lion will become drowsy, and will become vulnerable to being eaten by another hungry lion.  Any lion eating a lion gets drowsy, too.  
    In conisdering the fat of the solitary sheep, we are to assume that the best possible outcome for a lion would be to devour the sheep (or another lion) and thus satisfy his hunger; the next best outcome would be to remain hungry but remain alive; the worst outcome would be to eat the sheep (or another lion), only to be eaten in return.  And we must also assume that the lions in question behave very, very logically.
    What happens to the sheep?  

19.  There are five houses in a row (east to west), each of a different colour and inhabited by men of different nationalities, with different pets and preferences in beverages and cigarettes.
  1. The Englishman lives in the red house.
  2. The Spaniard owns the dog.
  3. Coffee is drunk in the green house.
  4. The Ukrainian drinks tea.
  5. The green house is east of the ivory house and next to it.
  6. The Old Gold smoker owns snails.
  7. Kools are smoked in the yellow house.
  8. Milk is drunk in the middle house.
  9. The Norwegian lives in the most westerly house.
  10. The man who smokes Chesterfields lives in the house next to the man with the fox.
  11. Kools are smoked in the house next to the house where the horse is kept.
  12. The Lucky Strike smoker drinks orange juice.
  13. The Japanese smokes Parliaments
  14. The Norwegian lives next to the blue house.
Who drinks water?  And who owns a Zebra?

20.  Archie, Brian and Joachim own a Ford, a Chevrolet and A Chryler, but not necessarily in that order.  One car is blue, one is green, and the third is brown.  Archie does not own the Ford, and his car is not blue.  If the Chevy does not belong to Archie, then it is green.  If the blue car is either the Ford or belongs to Brian, then the Chrysler is green.  If the Chevy is either green or brown, then Brian does not own the Ford.  Identify each person's car in terms of make and colour. 

21.  Bringing women into the orders of Toads, Foxes and Chameleons was a major leap into the '90s.  Toads always tell the truth, Foxes always lie, and Chameleons tell the truth and lie whenever they choose.  Four marriages among the clubs have everyone a bit confused.  We know that among the four couples, there is at least one man and one woman from each club.  Can you match the four men speaking below with their clubs, occupations (doctor, farmer, salesman, and teacher), wives (Rita, Sarah, Tina and Ursula), and wives' clubs? 

Leo
  1. No man married a woman from his own club.
  2. Nick is a Chameleon.
  3. Exactly two of the brides are Toads.
  4. Patrick isn't a Toad.
Melvin
  1. The teacher and Leo's wife are in different clubs.
  2. Leo is a Fox.
  3. Rita isn't a Fox.
  4. Patrick isn't in my club.
Nick
  1. Patrick isn't a Fox.
  2. Leo isn't in Patrick's club.
Patrick
  1. The salesman is a Fox.
  2. Ursula didn't marry the doctor.
  3. Leo didn't marry Tina.
  4. Melvin is in Nick's club.
  5. I am the farmer.
22.  Good poker players have four characteristics in common:  The are familiar with the odds associated with card distribution, they know when it is wise to bluff, they have poker faces, and they are lucky.  Angel invited four of his friend to play poker one evening, around his big circular table.  They were Babs, Cleo, Dot and Edie.  The following was also true:
  1. Everyone was sitting next to someone who knew the odds, but four of the five were sitting next to someone who was not well versed in the probabilistic aspects of the game.
  2. Four of the people were sitting next to wise bluffers, but three of them were sitting next to people who did not know when to bluff.
  3. Four of the people were sitting next to people with good poker faces, but everyone was sitting next to someone who could not keep a straight face.
  4. Exactly three of the people were sitting next to someone who was noted for good luck.
  5. Each of the players had at least one of the desirable traits; but only one, the big winner of the night, had all four.
  6. Babs knows the odds and knows when to bluff, but does not have a poker face and is not noted for luck.
  7. Edie is not sitting next to anyone who knows when to bluff.
  8. The person on Dot's right has a poker face.
  9. The person on Cleo's left does not know when to bluff.
  10. Angel knows the odds, but is not lucky.
Who is the big winner of the evening, what was the seating arrangement, and which traits did each of the five players have?

23.  The professor was a guest lecturer at a logic course for senior executives of major oil companies.  She selected six male students for this demonstration.  The professor placed fifteen dimes and fifteen nickels in six tin cups such that each cup contained the same number of coins, but a different amount of money.  She made six labels showing correctly how much money each cup held, but attached to each cup an incorrect label.  She explained the situation to the six students and gave a cup to each.  She asked each man in turn to feel the size of as many coins as he wanted in his own cup and announce something true about them.  The only evidence each man had was the size of the coins he felt, the incorrect label on his own cup, and the statements made by those who preceded him.  The first man said, "I feel four coins which are not all the same size;  I know that my fifty coin must be a dime."  The second man said, "I feel four coins which are all the same size; I know that my fifth coin must be a nickel."  The third man said, "I feel two coins, but I shall tell you nothing of their size; I know what my other three coins must be."  The fourth man said, "I feel one coin; I know what the other four coins must be."  Determine how the remaining two cups were labeled and what the total value of the money in those two cups was.

24.  When Maharaja Ram Singh died, he left 3465 gold pieces to be divided equally among his children.  Each wife had the same number of children and this number was 8 less than the number of wives per harem, which in turn was 4 more than the number of harems and 4 less than the number of gold pieces each child received.  How many children did Ram Singh have? 

25.  Comm. Dale E. Muter travels to and from work by train.  His train arrives at his hometown station at 6p each evening, and his wife always arrives promptly at this time to pick him up.  One day, Comm. Muter left work early and arrived at the station at 5p.  Not wanting to disturb his wife, he started to walk home along the route that she always drove.  When he was one-quarter of the way home, he met his wife.  They proceeded home at their usual driving speed and arrived home 12 minutes earlier than usual.  If the Muters live 12 miles from the station, how fast was Comm. Muter walking?

33.  Using the digits 1, 3, 4 and 6, form each of the numbers 22 through 30 using only the standard operations of addition, subtraction, multiplication, and division.  Here are two examples:  20 = (6 + 4) x (3 - 1), 21 = 31 - 6 - 4.  Here's a related question:  it's easy to express the number 9 using precisely three 3s if you're allowed plus signs:  9 = 3 + 3 + 3.  Now, using any mathematical symbols you like except plus signs, find three different ways to express the number 9 using preicsely three 3s and no other digits.  

34.  Identify the nine pairs of whole numbers (x, y) that satisfy:  xy/(x+ y)=10.  

35.  Thelma and Louise were standing on a street corner trying to decide which of two NYC restaurants to go to, Alfredo's or Bernardo's.  They eventually decided to eat at Alfredo's, because it was two blocks closer.  However, two crows were listening in on the entire conversation, and to them the decision made no sense; from the crows' point of view, Alfredo's and Bernardo's were exactly the same distance away.  
    Assuming the city blocks to be perfectly square, and assuming that neither restaurant was on the street or avenue that Thelma and Louise were already on, how far away were each of the two restaurants?  
    Can the situation ever be one step worse - that T&L think one restaurant is closer but the crows think the other one is closer?  Either provide an example when it is so, or prove that it is not possible.  

36.  It is easy to divide a square into four smaller squares.  It turns out to be impossible to divide a square into 2, 3 or 5 smaller squares, but all other subdivisions are possible.  Find a way to divide a square into six smaller squares.  Find a way to divide a square into seven smaller squares.   The explanations that 2 and 3 are impossible are pretty direct.  Can you explain why a square cannot be divided into five squares?  

37.  It is meaningless to say "This pencil is two longer than that pencil", but not to say "This pencil is twice as long as that pencil."  Explain why.  Not only is it meaningless to say "It is two warmer outside than inside", but it is also meaningless to say "It is twice as warm outside than it is inside."  Explain why this situation is different.

38.  Using the logical steps of arithmetic, find the number that each letter represents to form accurate long-division problems.  When completed, read the letters in order from 0 to 9 to spell a 10-letter word or phrase.   Digits are not repeated in each problem, but each problem is independent.

                CUR
IRE)RACKET
        RCSF
            AKE
             AIA
                TUT
                I RE
                RFA

                    LOW
IDOL)GOBLIN
            IDOL
              L I NB I
              LDNDO
                OGYRN
                OOOYW
                    I WGR

                 R IO
TEE)THROWS
        TCR I
           WSCW
              VVO
              I OHS
              I T I E
                 T I I


39.  For both of the following two puzzles, each triangle has a value from 1 to 9.  No two triangles have the same value.  The sums of certain groups of triangles are shown at their intersections.  Determine the value of each triangle. (I apologise for the vast amount of space here - if someone want to help me remove it, I'd be fine with that.  It's not important to me - I hope it doesn't bother you too much.)

if you can't see this image, come ask me about it.


More possible questions will be handed out in class during the first two weeks.