MAT115: Math for Liberal Arts

Assignment for Thursday, 1/14:

• Read "From Fish to Infinity" in your text (and make sure you watch the video from Sesame Street!)
• Examine that interesting Chinese artifact: what's going on?
• On our Canvas site:
  • Introduce yourself.
  • Participate in the discussion concerning this artifact on Canvas.

Try to fill in the next row of the Chinese version of Pascal's triangle, using bamboo counting rods. You might want to do it with Pascal's triangle first, so you'll know what numbers you're shooting for. See if you can discover how the system works.

Contribute to the discussion about The Great Fraudini. See if you can figure out the trick, and start reading other people's minds!

Homework due Monday, 1/25, at 8:00 am:

1. Write the following composite numbers as products of prime numbers:
   1. 33
   2. 32 (primes can be repeated!)
   3. 84

2. Imagine that you're playing Fraudini's trick with a friend: for each of the following numbers, indicate which cards contain that number by writing each number as a sum of powers of 2:
   1. 53
   2. 49
   3. 31

3. For the following use the method of "primitive counting" described in class:
   1. Turn the following into the appropriate string of 1s and 0s (drawing the tree for me is desired):
      1. 32
      2. 63
      3. 97

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

1. Please read this article on One-to-one correspondence.

2. In addition, you will begin work on your first IMath assignment. Please go to the IMath website and sign up (you can use your NKU username, but any password you like):
   • Course ID: 446
   • enrollment key: 3.14159

Due dates for IMath assignments are shown within IMath. (This one happens to be due on 2/5.)

And if you didn't do that reading from last time, how about doing it now?

1. Write the following numbers in the Mayan number system (assume places 1, 20, 360=18*20, 18*20*20, etc.), and in the Babylonian number system (using martini glasses and boomerangs):
   1. 58
   2. 223
   3. 818
   4. 7582
   5. 9433
   6. 79420
2. Complete the Mayan lunar calendar. Please show some work for the calculations -- no work, little credit.

1. Write a number as a sum of non-consecutive Fibonacci numbers;
2. Choose whether to go first or second in Fibonacci Nim, and then make your first move.

This time, however, there is a penalty for successive tries -- so you want to try to get things right on the first go!

1. "Working Your Quads" (p. 67 of our text)
2. Some readings (found here -- view each image separately for easier reading):
   • Background History of "Pascal"'s triangle
   • Properties of "Pascal"'s triangle

1. Assume that you have four different individuals, B, L, M, and A. Draw all possible distinctly different Facebooks with exactly two friendships. If you followed along in the video, you know exactly how many to expect, by looking in the proper row of Pascal's triangle.

   (Draw them as I did in the video.)

2. Demonstrate Egyptian multiplication by multiplying the following (write out the table, and check your work):
   1. 13*34
   2. 23*79
   3. 81*123
   4. 255*256

Homework (due 3/8 -- upload to Canvas): Demonstrate Egyptian division in two ways:

• 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. The Enemy of My Enemy (complete graphs)
2. Group Think (complete, directed graphs)

1. Draw the complete graphs with 6, 7, and 8 vertices. How many edges are there for each? What is the formula for the number of edges of a complete graph with n vertices?
2. Draw all the distinctly different simple graphs with one and two vertices (there aren't many!). How many were there for three, four, and five vertices (these were done in the "lecture", or were in the notes)? Can you find any pattern to the number of each? How many do you think there are with six vertices?
3. A floor plan for a house can be considered a graph: each room is a vertex, and each door between rooms is a "bridge" (an edge).
   1. Create a "floor plan" of your house (here's mine). Does it have an Euler path? Explain why it does or doesn't. (By the way, the "outdoors" is also a vertex! Your front/back door(s) leads to this "outdoors" region. In an apartment, out into the hall.)

      You can design your own floor plan, if you wish -- perhaps of your dream house. I'm not going to be checking that your floor plan is a faithful representation!

      If your house has multiple floors, pick your favorite.

   2. Add windows as additional edges (generally to the outdoors). Does that change things?

4. 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"). Four examples total.

1. Please read Chapter 27: Twist and Shout (Strogatz, p. 219)

2. Please check out these videos by Vi Hart:
   1. The first is called Snakes and Graphs. She's always got something really interesting to share.
   2. The second is called Mobius Music Box

3. To submit (due Thursday, 3/25):
   1. Check and verify Euler's formula for the Platonic solid graphs.

   2. Create your own Vi Hart squiggle (it's got to be closed), and show that it can be colored the way she does.

   3. Can you draw one of the knots she does, with snakes?

   4. Draw the borromean rings as snakes or ropes.

This reading highlights two important topics: the Reidemeister moves, and tri-colorability.

In the meantime, practice your five knots, and four links. Make sure that you can draw them, and distinguish them.

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.

You may use tricolorability as well as Reidemeister moves -- but you need to make it clear in each case how you're simplifying the knot until it's clear what the conclusion is. Redraw the knot each time you make a simplification.

Get started on these problems. We'll be talking more about these on Thursday, so don't panic!:)

Visit each of these two sites, and create your own fractal images (have fun!). Then post them in our discussion on Canvas. You might watch that third video from this day's materials to get some tips on doing this.

1. Fibonacci Spiral Fractal.
2. Mirror fractals.

1. On page 253, do you see that each time you slide down to the end of a diagonal you will have added the next triangular number of people into the hotel? I.e., 1, 3, 6, 10, ....

2. It would be nice to come up with a formula, so that we could shout out to all the buses exactly which room each person will be assigned, given their bus number and seat number. Can you find such a formula?

Please note: this is your final assignment, due this Thursday, 4/29. If you're late, you'll only get partial credit.

1. Submit your logo and a one-page paper as an assignment in Canvas.

2. Then put your logo and a description (you could just copy the text from your paper) onto a Canvas discussion board, so that everyone may enjoy them.

   It will be a digital poster session, essentially.