Math 22, Winter 2015, Home Page
Homework from the text:
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
Research Experiences for Undergraduates (REUs)
You may be interested in being part of a summer research project funded by the National Science Foundation, entitled Research Experiences for Undergraduates (REUs). Many of these provide stipends and support for travel and lodging. There are literally hundreds of such programs in a variety of science disciplines, including mathematics, and are a great opportunity to get experience in doing research. See this site for information. Most applications are due in February and March, but each sets its own deadlines.
Class 21, Tue., Mar. 17, 2015
We went over more on coloring problems and Dijstra's Algorithm.
We also learned about the doodling theorem (see Vi Hart's videos on doodling in math class.)
Here's a study guide for the final exam.
Class 20, Thu., Mar. 12, 2015
We went over coloring problems and Prufer's algorithm.
Class 19, Tue., Mar. 10, 2015
We learned about surfaces like toruses, and tried to embed K(5) and K(3,3) in the plane.
We learned about the Euler formula v – e + f =2 and realated formulas, and used them to show that K(5) and K(3,3) are non-planar.
We also learned about Euler circuits (each edge once) and Hamiltonian cycles (each vertex once), and saw how to
Eulerize" a street network by adding a minimal number of "double-back" streets to make all degrees even. We saw how to use "surgery" to find an Euler circuit in a connected graph with all even degree vertices.
We saw that two simple "greedy algorithms" (the nearest neighbor and cheapest link algorithms) for finding the minimal Hamiltonian cycle (the Traveling Salesperson Problem, or TSP) in a weighted graph might not work, and we might
need to use "brute force", which means examining every one of the (n-1)! cycles in K(n).
We saw some applications to circuit boards: using the Euler formula to find a lower bound for the thickness and thus the number of circuit boards needed to design a circuit graph, and how solving the TSP would simplify circuit board welding patterns.
Proving the TSP is NP-complete (or not!) would win you $1,000,000 and instant fame - it is the most important unsolved problem in this branch of mathematics.
We also reviewed Prufer's algorithm for codes for labeled trees and Kruskal's algorithm for finding a minimal spanning tree for a graph.
Extra homework: can you show how to embed K(7) in the torus with no edges crossing? Google it and you can find lots of images (if you give up!)
Class 18, Thu., Mar. 5, 2015
We went over material on generating functions, including looking at how the Fibonacci numbers may be set up as a generating function and thus the rational polynomial x/(x^2 + x –1).
We also briefly went over trees, including tree traversals inorder, preorder, and postorder, and their relationship to calculations.
Class 17, Tue, Mar. 3, 2015
We went over the exam and new material on 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.
Please turn in chapter 7 and 8 homework on Thursday of this week.
Class 16, Thu, Feb. 26, 2015
We had exam 2 and went over more on recursion relations.
Class 15, Tue, Feb. 24, 2015
We did class work on counting problems. We went over some modular arithmetic material including the Euclidean algorithm and solving
ax=b (mod m)
We went over material on recursions. Pick two starting numbers. Create a sequence in which the next number is the previous number minus the number previous to the previous, that is a(n) = a(n-1) - a(n-2). Solve and find a(60,000).
Class 14, Thu., Feb. 19, 2015
We went over recursion relations and learned about the Fibonacci numbers, including how they are found as the numbers of spirals on pine cones.
Here are some more links on Fibonacci numbers:
Vi Hart's Fibonacci number videos. (We watched the first of 3 in class.)
A
great
site
about
Fibonacci
numbers.
Fibonacci
site
with
lots
of
pictures
and
interactive
applets.
Interactive site
that helps explain phyllotaxis,
which is the pattern of spirals in many plants.
Pingala's possible use of the Fibonacci Numbers
Rachel Hall who credits Indian mathematician/musician with the Fibonacci numbers - see her article "Math for Poets and Drummers" listed at her site.
List of things to study for exam 2:
Calculations with factorials and binomial coefficients
Principle of inclusion exclusion examples
What makes something an equivalence relation
Counting problems
Permutations (counts where order matters)
Combinations (counts where order does not matter)
Numbers of subsets of different sizes
Solution so modular arithmetic problems of the form ax = b (mod m)
Number of ways to distribute n identical balls into k distinct boxes
Number of ways to rearrange the letters in a name, when some letters identical
Counting or combinatorial proofs
Solve linear homogeneous recursion relations of order two
Class 13, Tue., Feb. 17, 2015
We introduced "linear homogeneous recurrence relations of the second order" and the Fibonacci numbers (see chapter 8).
We also did an activity based on the permutahedron for four elements, and saw its connection to the musical form change-ringing, as well as to DNA analysis. I'll upload those slides for you shortly!
Please work on chapter 7 homework.
Class 12, Thu., Feb. 12, 2015
We went over chapter 6 and 7 material including the Principle of Inclusion/Exclusion (PIE).
Class 11, Tue., Feb. 10, 2015
We went over chapter 7 material.
Class 10, Thu., Feb. 5, 2015
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.
Class 9, Tue., Feb. 3, 2015
We went over the exam and and started chapter 6 on combinatorics. Please turn in chapter 3 & 4 homework on Thursday.
Class 8, Thu., Jan. 29, 2015
We had exam 1 and went over more from chapter 5.
Class 7, Tue., Jan. 27, 2015
We went over more from chapter 4 and also some from chapter 5.3 on modular arithmetic.
You have your first exam on Thursday; it will cover chapters 1-4.
Chapters 3 and 4 homework will be due next Tuesday.
We discussed more about Ramsey numbers, which were introduced when we played the "color the edges of K(6) game", which we proved cannot end in a draw.
Some things to know about for exam 1:
(1) Be able to draw a graph with a given set of degrees.
(2) Be able to represent a graph with a matrix and know how to use powers of the adjacency
matrix to find numbers of paths between vertices.
(3) Be able to decide whether two graphs are isomorphic.
(4) Understand the use of logical connectives, and be able to build the truth table for a logical
statement
(5) Understand the difference between a statement, its converse, its inverse, and its
contrapositive.
(6) Be able to give a simple proof by contradiction.
(7) Prove a statement by induction.
(8) Use the pigeonhole principle.
(9) Decide whether or not a function is one-to-one or onto.
(10) Use the multiplication and addition principles to solve counting problems.
(11) Decide how many functions there are from set A to set B.
(12) Use quantifiers for "for all" and "there exists" and their negatives.
(13) Use set operations like complement, cross product, union, intersection, and set difference.
Class 6, Thu., Jan. 22, 2015
We went over material from the end of chapter 3, and also induction proofs in chapter 4. What was wrong with the proof by induction that all cows are the same color?
Class 5, Tue., Jan. 20, 2015
We went over more graph theory, especially info on how to tell if one graph is the same or "isomorphic" to another.
We saw some examples of pairs of graphs with the same numbers of vertices, edges and degrees, but which are non-isomorphic.
We saw that if the edges of K(6) are colored blue and red then the pigeonhole principle guarantees that there will be either a red or a blue K(3).
Chapter 1 and 2 homework due this Thursday!
Class 4, Tue., Jan. 15, 2015
We learned about the "graph alphabet," and introduced graph theory.
Add to your homework the following: find two graphs with the same numbers of vertices, edges, and set of degrees, but which are different graphs. That is they are "non-isomorphic." There are plenty of examples with only 5 or 6 vertices. Your graph should be a "simple" graph with no loops or multiple edges.
We also found all 11 simple graphs with 4 vertices.
Because we still have one bit from chapter 2 to go over in class, I'm not going to take up chapters 1 and 2 homework until next Thursday, Jan. 22.
Here is that free graph and geometry software, Geogebra which might be useful in doing homework.
We found examples of functions from the integers Z to the integers Z which are
1 to 1 and onto
Not 1 to 1 and not onto
1 to 1 but not onto
Add to your homework: find a function from Z to Z
which is onto but not 1 to 1. You might try using the ceiling or floor function.
We went over more material on truth tables and also material on functions. 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 also learned about Russell's Paradox.
Last week 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 ....
Class 3, Tue., Jan. 13, 2015
The substitute, Kejian Shi, went over material on sets, logic, and truth tables.
Class 2, Thu., Jan. 8, 2015
Today we went over the pigeonhole principle. 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 they have no divisors other than 1 in common (they are "relatively prime.")
(5) A pair of your numbers had the property that one divided the other
equally.
(6) After striking out one of your numbers, two subsets of your numbers had the same sum.
We saw how the pigeonhole principle explained why properties 1-5 are true. Number 5 was tricky, and involved labeling the six pigeonholes with different numbers of numbers!
Property 6 is similar to an example in the text; in this case the we have to show that there are more possible subsets than there are possible sums.
The number of subsets of a six element set is 2 ^6 = 64, while the maximum sum of any such subset is 7+8+9+10+11+12 = 57, so two subsets must have the same sum.
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.
We also did the "handoff" problem, and noted that counting problems often involve making assumptions about exactly what is to be counted, then insuring that our count is consistent with our interpretation of the problem.
We learned a little more about modular arithmetic.
Here's a site on modular arithmetic.
Here's a site
which will do modular arithmetic calculations for you.
Here's a site
on modular arithmetic by Susan Addington.
Homework:
Read and work on homework from chapter 1. Chapters 1 and 2 homework will be due together when we have finished chapter 2. This will probably be on Tuesday, Jan. 20.
Class 1, Tue., Jan. 6, 2015
Here is the
pattern game we played in class.
We played the take-away game, in which each player removes 1 or 2 counters on each move, the last player to move winning. We developed a winning strategy involving leaving your opponent with a multiple of 3 on each turn, if possible.
We began learning about modular arithmetic.
Your first assignment is 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 the chapter 1.7 problems in the homework list above.
Links and notes from previous classes
Final Exam study questions.
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.
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.
We also learned about the 100 prisoners, 100 keys, and 100 boxes puzzle. See this link for an explanation, if you like!
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.
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.
You might want to read up on big oh notation. Here's another link.
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.
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!)
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.
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!!
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.