Assignments, MAT115

• Have a look at your text (The Joy of X), and pick out your favorite three sections. We can't cover everything, but we'll have a little time to make sure that we hit the class top 3. I'll collect your votes next time.
• Please read Probability (Idea #31, pages a and b)

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
   1. both come up heads?
   2. 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) six-sided dice. What is the probability you roll a sum of
   1. two?
   2. seven?
   3. fourteen?

3. Roll three ordinary, fair, six-sided dice.
   1. How many different outcomes are possible?
   2. What are the possible sums?
   3. What is the probability of rolling "triples" (all dice the same)?

4. 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. Review: Consider an ordinary, fair, six-sided die. What is the probability that
   1. two successive throws will give two different results (e.g., 1 and 3)?
   2. that three throws will give three different results?
   3. that six throws will give six different results?
   4. that seven throws will give seven different results?

2. In dividing seven different candies between two different children,
   1. in how many distinctly different ways can we divide the candies?
   2. what's the probability that we give one child three, and the other four?
   3. what's the probability that we give one child all, and the other none?
   4. What's the probability that one kid is unhappy?:)

3. Suppose that your company is testing for illegal drugs. Drug use is low (2%). The probability of testing positive for drugs given that one takes drugs is 80%. The probability of testing positive for drugs given that one doesn't take drugs is still 10%.
   1. What is the probability that one actually does illegal drugs given that one tests positive?
   2. What is the probability that one actually does illegal drugs given that one tests negative? (called a "false negative").

4. Monty Hall now has five doors, three cars, and two donkeys. You pick two doors, and if there's a car behind both, you win a car! What's the probability that you win a car by
   1. sticking?
   2. switching?
   Show all details!

1. Write the following numbers in both Babylonian and Mayan number systems:
   1. 57
   2. 222
   3. 817
   4. 9432
   5. 14449

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

3. Tap out your name (first and last) and birth year in Morse Code. Use this ("What hath God Wrought") as an example of how to do it.

4. 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

5. 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).

Homework #4: (due Wednesday, 2/18)

• Demonstrate Egyptian multiplication by multiplying:
  • 23*79
  • 81*123
  • 321*247
• 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 seven.

Homework (due Friday, 3/27):

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"

1. Draw the complete graph with 8 vertices. How many edges are there?
2. 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?
3. Check out this page and relate it to Euler's Konigsberg problem. Try some of the puzzles yourself.

Homework, due Wednesday, 4/22:

• Draw all simple graph on four distinguished, or labelled, vertices. Imagine that each one is a different person on Facebook.

• Draw your family tree (as a directed graph) with one of your grandparents as the root of the tree (include only those actually descended from that grandparent). Label each vertex (that is, each person!).

• Identify the knots (or links?) in this "story", which I call A Knotty Tale. You might use a piece of string to help you decide. Just put a name next to each one.

Homework (due next Wednesday):

• Use Knot Tiles on a piece of graph paper (such as the Mosaic Graph Paper) to draw your own 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

• Check out the PNC bank logo. Describe how its related to Mobius bands. Find another two logos for corporations that use mobius bands, or other knot-related ideas.