Math 22
Fall 2009 Home Page Green Sheet Class 21, Mon., Nov. 30, 2009 We went over a few homework problems. I emailed you all the quiz,
please do number 4 for homework.
If you did not receive the email, let me know and I'll send it out to
you individually.
Another counting problem I asked you to solve was to determine how many
4 by 4 Latin
Squares there are (in the context of the game KenKen. This turns out to be a
difficult problem - you can skip it!
We also spent time on first and second order linear recursions.
The additional homework problem was:
s(n) = 2s(n–2) – s(n–4), with S(0) = 0, s(1) = 1, s(2) = 2, and s(3) =
3. Find the solution (remember it will involve i and –i).
Do homework through section 9.3.
I will collect chapter 5 homework on Wednesday. Class 20, Wed., Nov. 25, 2009 We went over material from chapter 8.6, 9.1, and 9.2. Do homework
through the end of section 9.2.
Here's the 100 prisoner problem:
There
are 100 prisoners in cells numbered 1 to 100. The
keys, also numbered 1 to 100 are distributed among 100
boxes, also labeled 1 to 100. The prisoners are told
that each will be given the chance to open any 50 boxes,
looking for their own key. They may communicate
- and
strategize - before they start opening boxes, but no
communication is allowed after the first one starts
opening boxes. If all of them find their own keys,
then all will be released. What strategy do they adopt
in order to maximize their probability of success?
Class 19, Mon., Nov. 23, 2009 We went over part 2 of Exam 2.
We went over the 6 people at the party problem, an example of Ramsey Theory.
Homework: Prove that R(3,4) = 9. That is, if the edges of the graph K9
are colored red and blue, then there is either a red K3 or
a
blue
K4, or vice versa. Hint: see note at the
very bottom of this page!
We learned about combinations with repetition.
We began going over the law of inclusion exclusion, and will apply this
result to the problem of derangements,
also
known
as the hat-check problem.
Homework: work problems through section 8.5.
Try to work the problem below before looking at the link in the next
sentence!
We heard about the birthday problem: what is the "break-even" point
n for which the probability that n or
more randomly chosen people include at least two with the same
birthdate with probability more than 50%, and this is not true for
fewer than n people.
Class 18, Wed., Nov. 18, 2009 We went over the first part of exam 2, also some homework from
chapter 5, and the first three sections of chapter 8 (we will skip
chapters 6 and 7). Do homework from sections 5.5, 8.1, and 8.2. New problem to add to homework: If you choose 6 distinct numbers
from the set {1,2,3,...,10}, then one will be a divisor of another. Use
the pigeonhole principle to explain why this is true. Hint: odd numbers
and powers of 2; the number of elements labelling each pigeonhole will
not be the same - if you use a method that generalizes.
Class 17, Mon., Nov. 16, 2009 We went over material in sections 5.2 to 5.5. Do homework for
sections 5.2, 5.3, and 5.4.
We also heard about the doodling coloring theorem: doodle any number of
closed curves, so that curves may share common intersection points, but
not entire sections of the curve. The resulting diagram will be
2-colorable. Why? Prove this!
We also eard about the Travelling
Salesperson
Problem
(TSP), including the Nearest Neighbor Algorithm
and the Sorted Edges Algorithm (neither of which works all the time!)
Here's a site with links to
latest research.
Did you do the two extra problems below in class 16?
Class 16, Wed., Nov. 11, 2009 We went over sections 4.5 and 5.1, and also planar graphs.
By the way, here's a totally absorbing planarity game - move the
vertices so that the graph is drawn without edges crossing. Please try
playing the game ... but don't get addicted!
Two additional homework problems: (1)Can K(2,2,2) be drawn in the plane without edges
crossing? If so, do it!
(2) Draw the de
Bruijn graph for A,T,C,G which produces, via the Euler circuit, the
string of length 17 (called a "de Bruijn
sequence") which contains every possible substring of length 2. Do homework through section 5.1. Be ready to turn in chapter 4 homework.
Class 15, Mon., Nov. 9, 2009 Wehad exam 2 and also went over sections 4.3 and 4.4
We went over sections 4.1 and 4.2, and saw some applications of graph
theory.
Here's an
image of the internet graph.
Do homework for sections 4.1 and 4.2.
I've created a google group for this class for discussion purposes. I
emailed everyone registered in the class; if someone did not receive
the invitation, let me know and I'll add you directly.
Here's a study
guide
for
exam
2 on Monday, may be a few other topics on the exam
as well... (note this study guide is from a previous quarter, I'll try
to update shortly.)
Be prepared to turn in chapter 3 homework on Monday.
Class 13, Mon. Nov. 2, 2009 We completed section 3.6, and began the chapter on graph theory.
Do section 3.6 homework.
Class 12, Wed., Oct. 28, 2009 We went over some quantifier problems, also some more number
theory.
We also went over some material on codes from section 3.6; we'll finish
the coding material on Monday.
Your second exam has been put off until Monday, Nov. 9.
Class 11, Mon., Oct. 26, 2009 We spent the first hour going over number theory questions, as well
as some quantifier and logic questions.
The second hour was spent on code theory.
Begin homework for sections 3.4 and 3.5, we'll complete these sections
Wednesday, and cover 3.6 as well.
I'll take up the logic and quantifier homework on Wednesday.
Three prisoners enter a room and a red
or blue hat is placed on each person's head. The color of each hat is
determined randomly, for example 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 prisoners must simultaneously guess the color of
their own hats
or pass. The prisoners will all be set free if at least one
player guesses correctly and no players guess incorrectly. Otherwise
they receive life sentences.
What strategy should the players adopt to insure their success rate is
3/4? How is this related to Hamming codes? What if there are 7 players?
Class 10, Wed., Oct. 21, 2009 We spent much class time looking at examples and theory of the RSA
encryption algorithm.
Here is an article on Sarah
Flannery's
attempt
at
creating
an
alternative
to
RSA.
Finish the quiz we started on Wednesday, which is due Monday. Here is
an example
of
an
induction
proof
using
strong
induction.
Please read ahead the rest of chapter 3. We need to move through it
fairly quickly.
Please also be prepared to turn in the
logic problem handout and the handout
about
quantifiers on Monday.
If you have not read through the handout
on
modular
arithmetic, and worked the problems, assigned one week
ago, please do so now!
Here's the BOINC site, which
organizes volunteer distributed computing, and operates in the petaflop
range (quadrillions of operations per second.) BOINC stands for
Berkeley Open Infrastructure of Network Computing.
Class 9, Mon., Oct. 19, 2009 We went over exam, also went over some logic problems and methods
of proof, as well as quantifiers "for all" and "there exists."
Here are some more logic
problems, as well as problems on proofs that we looked at briefly
in class; please do for homework. Here is a site with about 20 proofs of the
irrationality of the square root of 2.
Please work the problems in the
logic problem handout due on Wednesday.
Also try the problems in handout
about
quantifiers. We'll go over these ideas more on Wednesday.
Also read this Fido
handout (originally
written for the math for teachers class) and work the two problems on
"bottlecaps."
Class 8, Wed., Oct. 14, 2009
We went over more of section 3.3, more on linear congruences, learned
how to apply the extended Euclidean algorithm to solve linear
congruences, and began the appendix (section A.1). Do homework from sections 3.1-3.3 and A.1 for Monday. Also read A.2
and A.3.
We've begun looking at logic. Here are some puzzles by Raymond Smullyan,
from
his
new
book,
Logical
Labyrinths. (Recent quote by Smullyan: "Why should I be worried
about dying? It's not going to happen in my lifetime!"):
An anthropologist visits an island where there are two kinds of people:
knights, who always tell the truth; and knaves, who always lie.
(1) The anthropologist comes across three inhabitants, A, B, and C. He
asks A, "Are you a knight or a knave?" A answers, but indistinctly so
the anthropologist can not understand what he says. He then asks B,
"What did A say?"B replies, "He said that he is a knave." At this
point, C piped up and said, "Dont' believe that; it's a lie." Was C a
knight or a knave?
(2) According to another version of the above story, the anthropologist
instead asks A how many of the three are knaves. Again A answers
indistictly, so he asks B what A has said. B then says that A has said
that exactly two are knaves. Then, as before, C says that B is lying.
Is it now possible to determine whether C is a knight or a knave?
(3) Next the anthropologist meets two inhabitants, D and E. D says,
"Both of us are knaves." What is D and what is E?
Work these problems for homework also.
Class 7, Mon., Oct. 12, 2009 Went over part of section 3.3 and rest of 3.2 (the Extended
Euclidean Algorithm), and had exam 1.
Class 6, Wed., Oct. 7, 2009 Went over sections 3.1 and 3.2
Class 5, Mon., Oct. 5, 2009 Your first exam is next Monday.
We went over basic set theory ideas, and why the basketball I brought
in is really a Venn diagram.
We went over basic ideas about relations that are posets or equivalence
relations.
We looked at functions that are all possible combinations of one-to-one
and onto (or not).
We examined inductive proof and looked at examples that are algebraic
as well as geometric.
Your homework: problems in sections 2.5 and 2.6 (see below) - I will
collect chapter 2 homework Monday.
Class 4, Wed., Sep. 30, 2009 We went over equivalence relations more, especially the concepts of
modular arithmetic.
We went over partial orders (section 2.3) and functions (section 2.4)
also.
Homework: text problems in sections 2.3 and 2.4 (see list below).
I also asked you to figure out what the "basketball" I brought to class
had to do with Venn diagrams.
And do this homework problem for next class too:
Let the domain and codomain be the set of all integers, Z. Find:
(1) A function that is one-to-one and onto.
(2) A function that is one-to-one but not onto.
(3) A function that is not one-to-one but is onto.
(4) A function that is not one-to-one and also not onto.
Class 3, Mon., Sep. 28, 2009 We went over many homework problems from sections 1.3 and 1.4, and
then
went over material from sections 2.1 and 2.2.
Do the homework for sections 2.1 and 2.2, but be prepared to turn in
homework
for 1.1-1.4. Here's a reference for the fast
exponentiation
algorithm we examined today.
The Wikipedia
write-up on
the empty set has an entertaining section on the philosophy of the
thing.
More on equivalence
relations.
In case you are wondering, wikipedia is usually pretty good on basic
mathematical
ideas.
Class 2, Wed., Sep. 23 2009 We went over material from sections 1.3 and 1.4.
Do homework from sections 1.3 and 1.4 (see list at bottom of this page).
Here's a reference on the Pythagorean
Theorem, which mentions its history before Pythagoras, including
visual proofs.
Here's more on Horner's
Method. See the note towards the end on its history.
We also talked about Russell's
paradox. This link has Russell's account of is development of the
paradox.