Math 22, Winter 2014, Home Page
Green Sheet
See the homework list below.
Class 21, Tue., Mar. 20, 2014
We reviewed and went over more graph theory and generating function material.
Class 20, Tue., Mar.183, 2014
We went over more graph theory material, also generating function material.
Class 19, Thu., Mar. 13, 2014
We went over more about graph theory, including spanning trees, and also learned about generating functions.
Please read this careful introduction to generating functions, Please add the following homework problems at the end of it to your homework:
1b, 2b, 3a, 4c, 11
Here is the Wikipedia entry on generating functions - more complete, but more terse.
Class 18, Tue., Mar. 11, 2014
We went over Euler circuits, Hamiltonian cycles, and Dijkstra's Algorithm for the shortest path between two vertices. We also learned how to Eulerize a connected graph so that all degrees are even, and tried the greedy algorithms Nearest Neighbor and Cheapest Link, as well as Brute Force, to solve the Traveling Salesperson Problem (TSP). We started on coloring problems in chapter 13.
Chapter 10 and 11 homework is due next Tuesday.
Class 17, Thu.,Mar. 6. 22, 2014
We went over more graph theory material from chapter 11 on Euler's formula relating numbers of vertices, edges and faces.
We also learned about the 100 prisoners, 100 keys, and 100 boxes puzzle. See this link for an explanation, if you like!
Class 16, Tue.,March 4, 2014
We went over exam answers, more about spanning tree and graph theory, and some recursion problems.
Turn in chapter 7 and 8 homework on Thuirsday of this week
.
The AMATYC exam is this Friday, consult the notes posted around math classrooms for details, see
http://www.amatyc.org/?page=SMLPastQuestions
for sample questions.
Class 15, Thu., Feb. 27, 2014
We had exam 2 and went over more about recursions.
Class 14, Tue., Feb. 25, 2014
We went over chapter 6 material on counting problems.
Class 13, Thu., Feb. 20, 2014
We went over questions about combinatoric or counting proofs. Here's the Wikipedia entry on combinatoric proofs, which has a few examples. Here are a few more examples. Here's a site with slides for most of the same counting proofs we've seen in class. Here's a counting explanation for one problem, in detail.
We also learned how to count the spirals on Fibonacci numbers, and saw how that sequence can be represented by a "second order linear homogeneous recursion relation." This is from chapter 8.
We saw Vi Hart's first two (of three) videos on Fibonacci numbers.
Your next exam was scheduled for Tuesday, but I've moved it back to Thursday of next week, Feb. 27, so we can finish chapter 7. Chapter 7 & 8 homework will be due following the exam on Thursday of next week.
Class 12, Tue., Feb. 18, 2014
We went over matieral from chapter 6 and 7, including the Principle of Inclusion/Exclusion.
Class 11, Thu., Feb. 13, 2014
We went over material from chapter 6 and began chapter 7. We also went over new material on modular arithmetic.
Class 10, Tue., Feb. 11, 2014
We went over more about modular arithmetic, also introduced binomial coefficients and counting concepts.
Class 9, Thu., Feb. 4, 2014
We went over material from chapter 4 on induction and also much of chapter 5. Please turn in chapter 3 and chapter 4 homework on Tuesday. You might want to read up on big oh notation. Here's another link.
Class 8, Tue., Feb. 4, 2014
We went over the exam, also the pile splitting induction problem.
Next class, Thursday, turn in your induction proof of the following: Start with a pile of n identical objects. At each step divide a pile with k > 1 objects into two smaller piles with r and s elements. At each such step calculate 1/r + 1/s. When all piles are of size 1 calculate the product of the all the above sums. Call the result f(n). Is f(n) an actual function (that is, does each n lead to only one calculated result?) Find a formula for this result, and prove that this formula is correct using induction.
Class 7, Thu., Jan. 29, 2014
We had exam 1, and also began going over induction, from chapter 4. Begin working on induction problems.
Class 6, Tue., Jan. 27
We went over more material on graphs, including Ramsey numbers (the fact that if the edges of K(6) are colored blue and red then there will either be a red C(3) or a blue C(3)). We also went over the representation of a graph by an adjacency matrix, and the fact that the nth power of that matrix shows the total numbers of paths of length n from one vertex to another.
I asked you to prove, as part of homework, that a 2-edge colored K(6) has at least 2 solid color C(3)s. (Hint: start with the fact that it must have one, proved in class!)
You might have a problem on the exam on representation of a graph by an adjacency matrix.
Exam 1 will be this Thursday, and will cover chapters 1,2, and 3.
Two of the logical connectives we learned about, that are not in the text, are NOR and NAND - you are responsible for knowing about these!
We learned about six degrees of separation, the Kevin Bacon game, and Erdos numbers last week.
I asked you to 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? (We answered this as no last class.)
Ramsey numbers and played the "color the edges of K(6) game, which we proved cannot end in a draw.
Here is free graph and geometry software, Geograbra.
Class 5, Thu., Jan. 23
We went over material through chapter 3.6.
Exam 1 will be next week and will cover chapters 1,2, and 3.
Bring a Scantron for the exam.
Please work on chapter 3 problems.
Class 4, Tue., Jan. 21
We went over chapter 3 material on functions and learned the graph theory alphabet. We found all eleven unlabeled "simple graphs" (no loops or multiple edges) with 4 vertices. I asked you to find a function from Z to Z that is onto but not 1 to 1.
Please turn in chapter 1 and 2 homework on Thursday.
Class 3, Th., Jan. 16
Sub went over chapter 2 material on logic and sets.
Class 2, Th., Jan. 9, 2014
We went over some of the text problems, also answered the pigeonhole principle questions.
We saw more about modular arithmetic, and did the mod 9 trick. The trick works because a number's digit sum is the same if the digits are rearranged. The digit sum is congruent to the number itself with respect to mod 9.I asked you to try to figure out why this is true. Here's a hint: rewrite the number as a "polynomial" in powers of 10. For instance, 357 = 3(10^2) + 5(10^1) + 7(10^0). Then convert to mod 9 ....
We learned why the number of subsets in a set A, with cardinality |A|=n is 2^n.
We also went over binary strings and the binary base system.
Your homework is to do the problems below in section 1.7. Remember, you will turn in homework at the end of chapter 2!!
Class 1, Tue., Jan. 7, 2014
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.
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 (they are "relatively prime.").
(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.
Homework
Chapter 1, section 1.7: 1,2,4,5,8-11,13-17,19,20,22-25.
Chapter 2 section 2.9: 2-6, 8-11,13,14,16,18,23,25.
Chapter 3, section 3.13: 2,4,5,6,7,11,12,14,16,17,24,26
Chapter 4, section 4.11: #5,10-14,17-19,21-23
Chapter 5, section 5.9: 2,4,5,6,8,9,11,14,19,20,2
Chapter 6, section 6.9: 2-5, Section 6.11: 1 and 2, Section 6.13: 1,3,7,8,10,16,15,17
Chapter 7, section 7.9: 1-8,10,11,14,17,19,21,24
Chapter 8, section 8.12: 1,3,6,7,8,11,16 (every other property starting with 1st), 19,22,25
Chapter 10, # 1-3,7-9,17-20,24,25
Chapter 11, section 11.10: 2,3,5,9-12,20-22,24
Chapter 12, # 5,6,8,10,15-17,23,24
Chapter 13, # 4,5,8,12,13
Links
Final Exam study questions.
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.
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.
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?
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.
We also discussed a tiling problem. Here is a site with a huge number of mathematical tiling results.
Here is that free graph and geometry software, Geograbra.
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}?
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?
We counted spirals on pine cones, saw Vi Hart's first (of three) videos on Fibonacci numbers.
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.