Assignments, MAT115

You might also check out my on-line explanation of the birthday problem, for comparison.

Consider playing "Let's Make a Deal" (so that you better understand it -- here's a good description of the game). Both of these problems are examples of Probability problems.

Put your name at the top of the paper and label it Math 115 - Homework 1. For each problem show me how you found the answer - unsupported answers are worth nothing. Feel free to explain your reasoning and, if you wish, you may include any incorrect attempts at the problem. If you use an outside source you should reference that source!

1. Flip a fair penny and a fair dime. What is the probability that

   a. both come up heads

   b. they come up with different results?

   Hint: If you find this difficult, you might want to list out the universe of all possible flips.

2. Roll two standard (fair) dice. What is the probability you roll a sum of

   a. two?

   b. seven?

   c. 14?

3. There are 5 people trapped in an elevator. Being really bored one of them bets the others that at least two of them were born on the same month of the year. What is the probability she wins this bet? ( You may assume that it is equally likely to be born on any given month. this may not be strictly true, but it is close enough that the final probability wouldn't change very much.)

1. Ifran, pages 1 and 2
2. Jacobs, pages 1, 2, 3, 4
3. Box and Scott

Problems A and B use the Jacobs reading from above, pages 1, 2, 3, 4

1. Use the pen diagram on p. 404 to do the following probability calculations:
   1. If you choose two pens at random, what is the probability that both of them would be green?
   2. If you choose two pens at random, what is the probability that one but not both of them would be medium?
   3. What is the probability that you choose three pens and two or more are the same?

2. Use the multiple-choice test diagram on p. 405 to answer questions 9 through 13 on that page.
3. 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. 97
      2. 63
      3. 32

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

• Mayan Math (Notice that the author says that we'll "skip the details" about the mixed base system -- we won't skip them!)
• Babylonian Math

1. Write the following numbers in both Babylonian and Mayan number systems:
   1. 57
   2. 222
   3. 817
   4. 9432

2. Complete the days of the Mayan lunar calendar (177, 354, etc.).

• 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. Rewrite the number we know as 2977 (written in base 10), only using
   • base 2
   • base 8
   • base 16
   • base 60

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).

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 seven.

1. Create the next row in the Chinese version of Pascal's triangle -- using the Chinese writing!
2. Find seven examples of rectangles in your daily life. Measure the side lengths, and compute the ratio of the larger to the smaller side. Rate them by how close they come to being golden.
3. In how many different combinations can 7 people get into two vans?
4. In how many different ways can 4 of the 7 people get into van A, and 3 into van B?

1. Please read this summary of Pascal's triangle.
2. Please read this summary of Platonic solids, to learn about the Platonic solids.
3. Cut and create Platonic solids out of paper, using this template. You may use these as a cheat sheet for the next exam. You must have put them together, however, and you must use only your own.

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

2. In your own words, explain why no Platonic solid has
   1. hexagonal faces
   2. octagonal 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. That is, a realistic view of a platonic solid on 2-dimensional paper. Try your hardest to do this well!

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?

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

• 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?
• Give two examples of balanced and two examples of unbalanced graphs with four people in them (see the first reading above).

• 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.

Homework (due Thursday, 11/20): Use Knot Tiles on a piece of graph paper (such as the Mosaic Graph Paper) to create knot-tile versions of

1. trefoil knot
2. figure-eight knot
3. cinquefoil knot
4. 5-twist knot
5. Borromean rings
6. Soloman's knot

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

   For the Mobius band cutting, you will want to use long and wide bands -- it makes seeing what's going on much easier.

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, and draw (or print) the symbols.