Assignments for Mathematics for Liberal Arts

Read Idea #31 -- Probability, p. 124-127, for Wednesday.

Consider the following three puzzles:

1. You are at a fork in a road, with two paths leading away. One path leads to heaven, while the other path leads to hell. In front of each path is a monster. One monster always tells the truth, while the other monster always tells a lie. You have no idea which monster is in front of which path. You can ask just one of the monsters just one question to correctly determine the path to heaven. What question do you ask? Remember: you don't know which monster is which....

2. Three missionaries and three cannibals arrive at a river, and all must reach the other side. A canoe sits, ready to carry them to the other side -- but it can only take up to two at once. If at any moment the cannibals outnumber the missionaries on a side of the river, they'll gobble up the missionaries. How can you get all six safely to the other side?

   Note: occupants in the canoe are not constrained to stay in the canoe, so if the missionaries are outnumbered on a side (including the canoe) the missionaries are toast.

3. You're incredibly thirsty, and luck upon a lemonade stand. A sign says "8 ounces of lemonade, $1.50", and you've got the cash. Unfortunately, the perverse lemonade salesman doesn't have any 8 ounce cups (only 6 and 10 ounce cups), so he refuses to sell you any (because he can't actually provide 8 ounces of lemonade). How can you convince him that he can actually dispense exactly 8 ounces of liquid?

Prepare a typed paragraph in response to each as you attempt to solve each of these. Diagrams or any other illustrations may be hand-drawn. This will be handed in (due Friday, 8/26). In what ways does probability figure into these problems?

Writing is important! Pay attention to spelling, grammar, and the clarity of your ideas.

• Read Idea 01, Zero, pp. 4-7
• Read Idea 09, Primes, pp. 36-39

1. What is the size of the sample space for the toss of a single coin? two coins? three coins? four coins? What pattern are you seeing? Guess the size of the sample space for the toss of 100 coins.

2. What's the probability of getting exactly two heads in four tosses of a fair coin?

3. What's the probability of getting a sum of 2, 3, or 12 in a roll of two ordinary, fair dice? What's the probability of getting a sum of 13? What's the probability of getting a sum less than 13? What's the probability of getting an odd sum?

4. What's the size of the sample space of a roll of three ordinary dice, one red, one white, one blue? What's the chance that the white die is less than 4, the red lands on 6, and the blue is even?

5. In the birthday problem, Sue is born in January, and Steve in March. What's the size of the sample space? What's the probability that Sue is born January 1st, and Steve is born March 1st?

1. Write the prime factorizations for
   1. 1008
   2. 1009
   3. 1010
2. Use the method of "counting by partition" to count the following (show the entire tree, and the string of ones and zeros that results):
   • 47
   • 337
   • 1008
3. Use the method of "counting by partition" backwards to report the number of sheep corresponding to the following strings:
   • 0, 0, 1, 0, 1, 0, 1
   • 1, 1, 0, 1, 0, 1
   • 1, 1
   • 0, 1
4. In the binary card trick (Fraudini's trick -- see this on-line description),
   • What number (from 1 to 63) is one thinking of if the cards chosen are the card with all the odds from 1 to 63, and the card with all the numbers from 32 to 63?
   • If I'm thinking of the number 42, describe which cards I'll choose.

• Demonstrate Egyptian multiplication by multiplying:
  • 23*79
  • 81*123
• Demonstrate Egyptian division by dividing:
  • 9/4
  • 13/7

Homework problems (due Friday, 10/7):

1. Rewrite the number we know as 2977 (in base 10), only using
   • base 2
   • base 8
   • base 16

2. Rewrite the following numbers in base 10:
   • 10100101001₂
   • 73556₈
   • DB92F₁₆

3. Show how to add 226₈ and 375₈ (both numbers expressed in base 8).

4. How would the Mayans write
   • 97
   • 162
   • 360

5. How would the Babylonians write
   • 97
   • 162
   • 360

6. Explain how Lewis Carroll was using unusual bases to do the strange math in chapter two of Alice in Wonderland? On about the third page of chapter two (after the graphic of "Giant Alice watching Rabbit run away"), Alice starts speaking out some bizarre equations:

   "Let me see: four times five is twelve, and four times six is thirteen, and four times seven is -- oh dear! I shall never get to twenty at that rate!"

   Find some reasonable bases to make the "calculations" work out....

Homework (due Monday, 10/10): I want you to visit this website and turn your own image into a Fibonacci spiral. Choose an appropriate image, then create and then email me a copy of your own spiral image. Make one that everyone will enjoy, as I will create a gallery of images.

Make sure that you send me the final image, and not just a link (that won't do anything!). Right click on the image, and save it to your computer -- then past the image into your email. That's the easiest way, I think.

• Fibonacci Nim:
  • Suppose you are about to begin a game of Fibonacci nim. You start with 50 sticks. What's your first move?
  • Suppose you are about to begin a game of Fibonacci nim. You start with 100 sticks. What's your first move?
  • Suppose you are about to begin a game of Fibonacci nim. You start with 500 sticks. What's your first move?
  • Suppose you begin a game of 15 sticks by taking 2; your friend takes 4; what's your next move, that will lead to victory provided you know the strategy?

• By experimenting with numerous examples in search of a pattern, determine a simple formula for

  

  that is, a formula for the sum of the squares of two consecutive Fibonacci numbers.

• By experimenting with numerous examples in search of a pattern, determine a simple formula for

  

  that is, a formula for the difference of the squares of two non-consecutive Fibonacci numbers.

• The rabbits rest. Suppose we have a pair of baby rabbits -- one male and one female. As before, the rabbits cannot reproduce until they are one month old. Once they start reproducing, they produce a pair of bunnies (one of each sex) each month. Now, however, let us assume that each pair dies after three months, immediately after giving birth. Create a chart showing how many pairs we have after each month from the start through month seven.

1. Explain how this image (of Earth...) is related to Platonic solids.

2. In your own words, explain why no Platonic solid has hexagonal faces.

3. Find an example of a company's logo which involves Platonic solids (don't use those you find using these resources, but they'll get you started):
   • Logos!
   • (explain how this one is related to Platonic solids)

4. Draw 2-dimensional projections of each of the Platonic solids.

5. For each of the Platonic solids, compute the following: where F is the number of faces, E the number of edges, and V the number of vertices. What do you discover?

6. Find a soccer ball and try the same thing on that: what do you discover?

7. Draw all the distinct simple graphs with five vertices (by hand). Make sure that you have them all! How many are there? How many are there of each "type", given by the number of edges? How can you use symmetry to help you find all the graphs? How can you use duality?
8. Draw your family tree with one of your grandparents as the root of the tree (include only those actually descended from that grandparent).
9. Here is another situation that Euler considered: Is the new, improved Konigsberg walk possible? Draw the corresponding graph and explain!

• Identify the knots (or links?) in this "story", which I call A Knotty Tale. You may need to apply the Reidemeister moves to convince yourself that a picture of a knot is really the unknot, say, but you don't need to tell me how you determined which knot or link each one is. Just put a name next to each one.

• Starting with the knot
  use the Reidemeister moves I, II, and III to create the "alternate square depiction"
  Please draw carefully on a blank, unlined page. Make your moves one at a time, and label each move.

• Check out the Rolfsen Knot Table, and decide if the knots , , , , , are tricolorable.

• a summary of some knot concepts.
• Please read Idea #23, Topology (p. 92)

Homework assignment, for Friday:

1. Twist a band in two different ways:
   1. four times, and
   2. five times,
   and cut the band in "two". What objects result? Describe them exactly, specifying their number of half twists, how they're connected, etc.

2. Describe exactly what you get if you cut a thrice-twisted band in thirds (as we did in class to the Mobius band).

3. Relate the following logo to twisted bands (e.g. Mobius bands):

4. Is the following recycling symbol correct (i.e. Mobius) or not?

5. Find two examples of the recycling symbol on nationally known products, one Mobius and the other not Mobius. Name the products.

1. a stick fractal, and
2. a collage fractal.

• Define the simple rule (e.g. how does a stick turn into other sticks?)
• Apply the rule at least twice, so that we can begin to see "the world within the world"