Math 22, Spring 2012, Home Page
Class 21, Wed., June 20, 2012
We went over chapter 13, and heard more about error-correcting codes.
Chapter 12 homework: # 5,6,8,10,15-17,23,24
Chapter 13 homework: # 4,5,8,12,13
Homework from Daxi's handout: 1,2,5,9,15,19
Homework from generating functions homework: problem at the end!
This homework due during the final exam.
Class 20, Mon., June 18, 2012
We went over material from chapter 12, and heard about error-correcting codes.
Class 19, Wed., June 13, 2012
We went over chapter 11 on the Euler formula and planarity.
We also went over breadth first and depth first searches.
We did an activity about mobius strips.
Chapter 11 homework, section 11.10: 2,3,5,9-12,20-22,24 (we'll go over some of these Monday.)
Chapter 10 and 11 homework is due next Wednesday. Chapter 12 will be due during the final exam.
Class 18, Wed., June 11, 2012
We went over chapter 10 material.
Class 17, Wed., June 6, 2012
We went over more of chapter 7 on recursion, and began chapter 10.
Homework for chapters 7 and 8 is due on Monday, June 11.
We are skipping chapter 9.
Homework for chapter 10: 1-3,7-9,17-20,24,25
Class 16, Mon., June 4, 2012
We went over chapter 8 material.
Ch. 8 HW, section 8.12: 1,3,6,7,8,11,16(every other property starting with 1st), 19,22,25
We also got exam part 1 back, and saw a parity magic trick, and learned why the probability of a derangement is 1/e when "reading" card names.
We also had an initial look at generating functions.
Here's a nice introduction to generating functions.
Class 15, Wed., May 30, 2012
We had exam 2 and began chapter 8 on linear recursions.
We counted spirals on pine cones, saw Vi Hart's first (of three) videos on Fibonacci numbers. (Watch the rest on your own!)
Class 14, Wed., May 23, 2012
We went over chapter 7 material.
Exam 2 will be on Wednesday on chapters 4,5,6, and 7; bring scantron.
Chapter 5 and 6 homework is due on Wednesday following exam.
Class 13, Mon., May 21, 2012
We learned about "change-ringing" church bells, and the relation to Hamiltonian cycle problems and DNA sequencing.
We worked on proofs. You have the proof handout due Wednesday.
We also worked on chapter 7 problems.
Ch. 7 HW, section 7.9: 1-8,10,11,14,17,19,21,24
Class 12, Wed., May 16, 2012
We went over most of the rest of chapter 6.
Chapter 5 and 6 homework will be due next Wednesday.
We also started on the "4 in a row" problem. Two permutations of four people, A,B,C, and D, are linked by an edge if they differ by one switch of two adjacent neighbors. So ABCD is linked to BACD, ABDC, and ACBD, etc. Find a list of all 4! permutations such that any pair in the list are linked by an edge, no permutation appears more than once, and the list ends where it started. To help you I've created a Geogebra file with most of the edges already added in, and will email to you. Find Geogebra here. Please prepare the graph of the 24 permutations by altering the Geogebra file so that all edges are present, and so that the graph is symmetric and "pretty." Once you've done this it will be easy to find a solution! Hand in the graph along with your solution on Monday.
Class 11, Mon., May 14, 2012
We went over the induction proof for the Splitting pile puzzle, and saw a visual proof too.
We also did a handout on star polygons and saw a number of images from various cultures.
We had a brief intro to binary codes.
We started chapter 6.
Chapter 6 homework:
Section 6.9: 2-5
Section 6.11: 1 and 2
Section 6.13: 1,3,7,8,10,16,15,17
Class 10, Wed., May 9, 2012
We went over the rest of chapter 5, including some material on modular arithmetic.
Here is the puzzle that uses the idea of error-correcting codes:
Three players enter a room and a red or blue hat is placed on each person's head. The color of each hat is determined by a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players' hats but not his own.
No communication of any sort is allowed, except for an initial strategy session before the game begins. Once they have had a chance to look at the other hats, the players must simultaneously guess the color of their own hats or pass. The group shares a hypothetical $3 million prize if at least one player guesses correctly and no players guess incorrectly.
What strategy should the players adopt?
Here are the seven questions from the "magic trick" that also uses Hamming error-correcting codes:
Think of a number from 0 to 15
Answer the following 7 questions with Yes or No, and write down your answers in order. You are allowed to lie at most once:
1. Is the number 8 or greater?
2. Is it in the set {4,5,6,7,12,13,14,15}?
3. Is it in the set {2,3,6,7,10,11,14,15}?
4. Is it odd?
5. Is it in the set {1,2,4,7,9,10,12,15}?
6. Is it in the set {1,2,5,6,8,11,12,15}?
7. Is it in the set {1,3,4,6,8,10,13,15}?
Homework section 5.9: 2,4,5,6,8,9,11,14,19,20,24
Here are sarah-marie belcastro's Geogebra files, for use with any graph theory problems.
Here is that free graph and geometry software, Geograbra.
Class 9, Mon., May 7, 2012
We went over new material from chapter 5 on modular arithmetic.
We also discussed a tiling problem. Here is a site with a huge number of mathematical tiling results.
Chapter 3 and 4 homework due this Wednesday.
Class 8, Wed., May 2, 2012
We had exam 1 and went over more of chapter 4.
Chapter 3 and 4 homework will be due next Wednesday.
We went over induction proofs.
Please complete the problem we worked on Monday: Start with a pile of n objects; split them into two piles with r and s objects and multiply r times s.
Continue this process until you have n piles with one object in each pile. Add all the products together. Prove that the sum is the n+1st triangular number.
Ch. 4 homework:
Section 4.11: #5,10-14,17-19,21-23
Class 7, Mon., Apr. 30, 2012
We went over more of chapter 3, and started chapter 4 on induction proofs.
By the way, should we even be giving "exams?" - see this article "Stop Tellng Students to Study for Exams, from the Chronicle of Higher Education!
Class 6, Wed., Apr. 25, 2012
We went over more of ch. 3 and some problems from ch. 2. We played Brussel Sprouts and I asked you to figure out why it was a completely determined game (meaning if the number of crosses you start with determines the total number of moves in the game.) Try to figure out why. If you want to google it you'll find lots of explanations, but don't do that till you've tried it!
We also learned about Ramsey numbers and played the "color the edges of K(6) game, which we proved cannot end in a draw.
Here is that free graph and geometry software, Geograbra.
Exam 1 will be on Wed., May 2, not Monday, and be on chapters 1,2, and 3. It will have scantron questions and a few non-scantron questions.
Homework to turn in, from section 3.13:
2,4,5,6,7,11,12,14,16,17,24,26
Class 5, Mon, Apr. 23, 2012
We learned the graph alphabet! - and started chapter 3.
We learned about six degrees of separation, the Kevin Bacon game, and Erdos numbers.
Find an onto function which is not one-to-one, which maps the integers Z onto Z.
Can you draw the complete bipartite graph K(3,3) on paper without the edges crossing?
Homework, chapter 3:
Do the "Try these," but do not turn them in.
To turn in, from section 3.13:
2,4,5,6,7,11,12,14,16,17,24,26
Class 4, Wed, Apr. 18, 2012
We went over the rest of chapter 2, including predicate logic (universal and existential quantifiers)
and more about propositional logic.
Turn in chapter 1 and 2 homework on Monday.
Class 3, Mon, Apr. 16, 2012
We went over material from chapter 2, as well as problems 24 and 25 in chapter 1.
We learned why the standard basketball has the pattern of a Venn diagram for 3 sets.
We learned to play the game Set.
Here's a handout about the Fido puzzle from class: Where's
Fido? Add the bottlecaps puzzle at the end of the handout to your homework.
Two of the logical connectives we learned about, that are not in the text, are NOR and NAND.
We also learned about Russell's Paradox.
In the pigeonhole principle "magic trick," I asked you to choose seven
numbers from the list 1,2,3,...,12. The properties each of your lists
had were:
(1) A pair of your numbers had a sum of 13.
(2) A pair of your numbers had a difference of 6.
(3) A pair of your numbers had a difference of 3.
(4) A pair of your numbers had the property that their only common
factor was 1.
(5) A pair of your numbers had the property that one divided the other
equally.
We saw how the pigeonhole principle explained why properties 1 and 2
are true. Can you use the pigeonhole principle to explain
properties 3 and 4? Remember, it's all in how you label to (six) pigeonholes!
Property 5 is more difficult to explain; in this case the six
pigeonholes have different numbers of numbers assigned to them! For example, one pigeonhole would have 3,6, and 12.
Chapter 2 homework, section 2.9: 2-6, 8-11,13,14,16,18,23,25.
Class 2, Wed., April 11, 2012
We finished going over chapter 1 problems, including some from homework. Read sections 2.1 - 2.3, and do the "Check Yourself" problems at the end of sections 2.2 and 2.3. The answers to the "Check Yourself" problems are in the back of the book - but try them first!
We also learned about various base systems, including binary and applications of binary in measurement systems, duodecimal (base twelve), and base sixty. To see some of the ideas about the use of binary systems in ancient Africa, see Ron Eglash's Ted Talk.
For history relating use of binary in India and China (as in the I Ching), see the Wikipedia entry on binary. See also a note on the use of octal systems by Native Americans, and the proposal 200 years ago for octal rather than base ten metric system at this page.
Class 1, Mon., April 9, 2012
We went over problems from section 1.2 of the text.
Your first assignment was to read chapter one and work the “Check Yourself” problems as
you read (not to be turned in, answers are in the back). As part of the homework that you will
turn in, do these problems in section 1.7: 1,2,4,5,8-11,13-17,19,20,22-25. Begin reading chapter 2.
We went over the sum, multiplication, and pigeonhole principles.
In the pigeonhole principle (PHP) "magic trick," I asked you to choose seven
numbers from the list 1,2,3,...,12. The properties each of your lists
had were:
(1) A pair of your numbers had a sum of 13.
(2) A pair of your numbers had a difference of 6.
(3) A pair of your numbers had a difference of 3.
(4) A pair of your numbers had the property that their only common
factor was 1.
(5) A pair of your numbers had the property that one divided the other
equally.
We saw how the pigeonhole principle explained why properties 1,2, and 3
are true. On Thursday we saw how to use the pigeonhole principle to explain
property 4. Remember, it's all in how you label to (six) pigeonholes!
Property 5 is more difficult to explain; in this case the six
pigeonholes have different numbers of numbers assigned to them! For example, one would have 3,6, and 12.
We also saw a geometric version of the PHP, used to show why 5 Starbucks located in a 2 mile by 2 mile square must include two that are within 1.5 miles of each other.