Plan and Assignments, MAT115-006, Fall, 2010

Read Idea #33 -- The birthday problem, p. 132-135.

Consider the following three puzzles:

1. You are at a fork in a road, with two paths leading away. One leads to heaven, while the other leads to hell. In front of each path is a monster. One always tells the truth, while the other always tells a lie. You have no idea which monster is in front of which path. You can ask one of the monsters 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 need to reach the other side. A canoe sits, ready to carry them to the other side -- but it can only take two. 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?

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 for next time. Diagrams or any other illustrations may be hand-drawn. This will be handed in. Writing is important! Pay attention to spelling, grammar, and the clarity of your ideas.

1. What is the biggest number that you can express with your two hands (assuming 10 fingers)?
2. What does each of the digits in the number 1729 mean?

Due Tuesday, 9/7: type up paragraphs (or do the follow calculations) in response to the following questions or problems:

• How was Lewis Carroll using bases to do the 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....

• Read the on-line Bases reference, and do the exercise at the bottom: "How do you add numbers in base eight? Of course, you do it the same as base ten, but you have to be careful: for instance, "6+4=12" in base eight! Make up and try some problems [try four different ones], and check your answers by converting everything to base ten."

  In particular, show how to add 147 and 173 (both numbers expressed in base 8).

• Rewrite the number we know as 1729 (in base 10), only using base 2.

• Use Egyptian mathematics to
  • Multiply 27 by 82.
  • Divide 5 by 8.
  • Divide 5 by 9.
• Write the first 16 numbers in base 8, as though you were counting.
• We can write binary "decimals" ("binimals?"), by adding places for 1/2, 1/4, 1/8, 1/16, etc. Demonstrate how to add 10110101.101 and 100101.01 in base two.

1. Check the Fibonaccis: starting from the first two Fibonacci numbers 1 and 1, and using the recursive formula, compute the first 21 Fibonacci numbers.
2. 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.

3. 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.

4. Fibonacci Nim: Suppose you are about to begin a game of Fibonacci nim. You start with 50 sticks. What's your first move?
5. Fibonacci Nim: Suppose you are about to begin a game of Fibonacci nim. You start with 100 sticks. What's your first move?
6. Fibonacci Nim: Suppose you are about to begin a game of Fibonacci nim. You start with 500 sticks. What's your first move?
7. Fibonacci Nim: 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?
8. 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 nine.

1. Find the prime factorization of the following:
   • 1763
   • 30030
   • 997
2. What is the golden ratio? How does it relate to the Fibonacci numbers?
3. Draw a really beautiful Fibonacci spiral, using graph paper and colors. You can make your own graph paper at this site.
4. Describe how the Fibonacci spiral relates to the golden mean.
5. Visit this website and generate an image in a Fibonacci spiral: web interface. Email me a copy of your final spiral image!

1. Visit this Wikipedia site to learn more about the platonic solids.
2. Complete your set of paper platonic solids. You'll be able to bring those to the next exam! If you need another sheet, you can can get one here: Platonic Solids in 2-D (paper template)
3. Think of another application of platonic solids (besides a calendar for the dodecahedron!).
4. Attempt to sketch drawings of the tetrahedron, cube, and octahedron from various perspectives.

1. 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?
2. Find a soccer ball and try the same thing on that: what do you discover?
3. Suppose you have a cube with sides of length 1. Suppose you construct an octahedron inside it whose vertices are in the centers of the faces of the cube. How long are the edges of that octahedron?
4. Suppose you have a cube with sides of length 1. Suppose you construct an octahedron around it by making the center of each triangle of the octahedron hit at a vertex of the cube. How long are the edges of the octahedron?
5. Report on another application of platonic solids (besides a calendar for the dodecahedron, or any other application we've already discussed).

Problems, due Thursday, 10/21:

1. 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?
2. Draw your family tree with one of your grandparents as the root of the tree (include only those actually descended from that grandparent).
3. Make your own (attractive, hand-drawn -- although you may use mosaic tiles) "Knots and Links" display chart, on a single page, illustrating the following knots and links on one 8.5x11 sheet of paper:
   • Borromean rings
   • Trefoil knot
   • Figure eight knot
   • Both five knots
4. There are two knots with five crossings. Demonstrate that they're different.

1. Try my funhouse mirror generator: make your own funhouse mirror image, using my web interface and your own image, and I'll post them on our "website gallery".
2. Create your own fractal: create a process/image/graph whatever that has fractal properties. It should not be something you merely found on the internet, but should possess some feature unique to yourself.
3. Write up (one page) an idea you have for your project. Do not write "I have no idea."

1. The problems with voting methods
   1. What voting system do we generally use in the United States?
   2. What can go wrong? Could you explain the example to a friend?
   3. What are some of the suggested alternative voting systems, and what are their advantages over our usual voting system?
   (Here's another similar reference, for your information.)
2. Schooling the Vote
   1. How could one use the results of this study for good, or for evil?
3. Ballot Roulette
   1. What are some of the worst failures of technology in voting history?
   2. Which do you think is the best system for preventing "spoiled" elections?

1. A picture of you at your precinct, with your voting container;
2. A written description of your experience (one page, typed, submitted to me via email -- to be shared with the class through our homepage, so make sure you don't include anything you wouldn't want out on the internet)
3. The results at your precinct. Include
   • Time of day (start, end)
   • The number of voters you contacted (include all voters you spoke with, including those who refused to answer), and the tally for each candidate.

For Tuesday, 11/9: revise your project idea (make sure you type it, or you will be losing points). Email it to me by Tuesday.

1. Consider the iterated map Run 10 steps, with , , , and each time starting from the value How would you characterize the long term behavior (if we keep doing this) in each case? What would happen in each case if we continued iterating forever?

   You might use the on-line animation to do this exercise.

2. Use the Excel spreadsheet, and make small changes in the initial value with . Describe what happens to the first few iterates, compared to the last few iterates. You can describe how the pattern of dots changes in the scatterplot. Find a small change that doesn't seem to affect any dots, and a small change that seems to affect most of the dots.

3. Use the logistic handout, , and cobweb starting from two different values of the initial value (the sheets show the choice ). Do at least four iterations.

• What sorts of patterns of cells that will lead to "extinction" (no cells alive)?
• What sorts will lead to a stable (unchanging) universe?
• What sorts will lead to a repeating universe (in the sense that the pattern of live cells will eventually appear the same as the initial universe, but not be static and unchanging)?
• What are some that will grow larger and larger (i.e. more cells alive) as time goes on?