MAT115: Math for Liberal Arts

Your first homework, to turn in: on your handout, fill in the next row of the Chinese version of Pascal's triangle, using the notation of the bamboo counting rods. You might want to do it with Pascal's triangle first (on the other side), so you'll know what numbers you're shooting for. This will be due next Tuesday. Include an explanation of how you chose to represent any numbers that haven't already appeared in the table.

1. Use Fraudini's trick to write the following numbers as sums of powers of 2 (you'll need some additional powers of 2):
   1. 31
   2. 57
   3. 129
   4. 222
   5. 817

2. For the following use the method of "primitive counting" we studied on day 3
   1. Turn the following into the appropriate string of 1s and 0s (drawing the tree for me is best):
      1. 32
      2. 63
      3. 97

   2. Turn the following strings of 1s and 0s into the appropriate number (again, drawing the tree for me is best):
      1. 101010
      2. 1010101
      3. 10110001

3. Write the following numbers in the Babylonian number system:
   1. 57
   2. 222
   3. 817
   4. 9432
   5. 14449

4. Extra credit: write a one-page story about the kid who created the Babylonian clay tablet nines table we studied, and about how the tablet ended up in our hands today. Complete fiction appreciated. (I'll post these, and we'll have a contest -- the winner(s) will win "get out of homework free cards" as well).

• Write the following numbers in the Mayan number system:
  1. 57
  2. 222
  3. 817
  4. 9432
• Demonstrate Egyptian multiplication by multiplying:
  • 23*79
  • 81*123
• Demonstrate Egyptian division by dividing:
  • 9/4
  • 13/7
  Try these using the same sort of "doubling/halving" table that we use for multiplication.
• Demonstrate Egyptian division by dividing:
  • 4/9
  • 7/13
  Try these using the unit fractions table method, and Fraudini's trick (writing a number as a sum of distinct powers of 2).

1. Fibonacci Nim:
   1. Suppose you are about to begin a game of Fibonacci nim. You start with 50 sticks. What's your first move?
   2. Suppose you are about to begin a game of Fibonacci nim. You start with 100 sticks. What's your first move?
   3. Suppose you are about to begin a game of Fibonacci nim. You start with 500 sticks. What's your first move?
   4. 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?

2. 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.

3. The rabbits rest. Suppose we have a pair of baby rabbits -- one male and one female. As before, a pair cannot reproduce until they are one month old. Once they start reproducing, they produce a pair of bunnies (one bunny of each sex) each month. This new pair will do the same as the parent pair -- mature, and reproduce following the same rules. 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 six.

You must print off and put your answers on a copy of the knots (or else draw them meticulously by hand). Otherwise it's a zero. No exceptions.

1. The Enemy of My Enemy (complete graphs)
2. Untangling the Web (directed graphs)
3. Group Think (complete, directed graphs)

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

2. 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)

3. Draw 2-dimensional projections of each of the Platonic solids. That is, a realistic view of a platonic solid on 2-dimensional paper. Try your hardest to do this well!

4. 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?

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

• Draw the complete graphs with 6, 7, and 8 vertices. How many edges are there for each? Can you figure out a formula for the number of edges of a complete graph with n vertices?
• Draw all the distinctly different simple graphs with five vertices (There are a lot! How many?). Use symmetry as much as you can to avoid double counting them. Can you see any patterns in how they're created? Which are duals to each other?
• Create "floor plans" of a house that has an Euler path, and one that doesn't. Explain why they do or don't.
• Give two examples of balanced and two examples of unbalanced graphs with four people in them (see "The Enemy of my Enemy is my Friend").

Homework:

1. Try these problems.
2. Create your own examples of
   1. a stick fractal, and
   2. an area fractal.
   You'll need to
   • 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"