Math 44, Spring 2008, Home Page

Math 44, Spring 2008
Home Page

Green Sheet

Here's that review sheet. I emailed it this afternoon because this site was down again.

Class 22, Thu., June 19
We heard a few more project reports, and reviewed some likely exam problems.
I'll post more of a study guide shortly.

Class 21, Tue., June 17
We saw lots of project reports.

Class 20, Thu., June 12
For those who made it, we had a wonderful visit with Dale Seymour at his magnificently outrageous home!
We owe Dale a real thank you!
Turn in the handouts (see below) on Tuesday. Projects due Tuesday also

Class 19, Tue., June 10
We heard reports. We spent some time on projects and on organizing rides to Dale Seymour's house on Thursday. Be there by 1:40, it is only a few miles from campus! We will end class in plenty of time for you to make the next class period, if necessary. Please do not be late.
Dale's address:
11170 Mora Drive.
Los Altos, CA 94024
Here's a link to a map.
We also constructed the five platonic solids with loops of rope.

I handed out a page of fractals; calculate the dimension of each and prepare to turn in next class (along with Eulerization problem handed out last class.)

Class 18, Th., June 5
We spent time working on the final projects, which are due during the last week of class (Tuesday of the last week.)
Please turn in your final project form this coming Tuesday.

We also went over more material on fractals.

I also gave you a sheet with a problem on Eulerization to return on Tuesday.

Your paper on a chapter or two from a book related to this class is due next class, Tuesday, June 10. The paper should be a 600 word summary of what you learned, how it is related to the class, and what about it intrigued you. I may ask you to give a brief (2 minute) oral summary on Tuesday.

New homework:
HW: Ch. 6.4 #2,3,6,8,12-15

If you can read Spanish, here's an article in the Costa Rican national newspaper about the conference I attended last week (some work I did with students there is mentioned in the article).

Here's that link to a recent online story about the math/dance work I do (under "math and computers, click on "Do the Math Dance," it has a short video clip).

Class 17, Tue., June 3
Here is your final project form; fill out as best you can, and bring to class on Thursday. (Ignore the old due date - that's from a previous quarter!)
We went over material from 6.3, 6.4, and 6.6.
Please do the homework from 6.3 and 6.6:
HW: Ch. 6.3 #1,2,4,5,9,13,21,23
HW: Ch. 6.6 #1-6,10-12,15

Class 16, Th., May 29
Bambi Moise subbed today.
Saw video and also worked on handout on cellular automata

Class 15, Tue., May 27
Bambi Moise subbed today.

Here is a message I emailed today:

The sub told me that you all thought the take-home was due next Tuesday instead of this Thursday. That's OK, but then the next paper is due one week after, and the project is due one week after that....

Also, she said that "Problem 3 , part a: m and n are not defined." In the example above part (a) there is an example of a cylindrical 3 by 4 street grid, which has 3 rows and 4 "columns" of vertices. A (cylindrical) m by n street grid has m rows and n columns arranged in the same cylindrical fashion. By the way, to work on this problem, construct a bunch of examples with small values of m and n, and gather data until you see some overall patterns!

And here's another message sent today:
Hannah sent me a detailed email with two questions. (Thank you Hannah!).

I am out of the country and will be back in class next Tuesday (when the take-homes are due). I cannot push the due date back further than that. If you have questions, email me and I'll respond to the whole class. As I've said, the most difficult part of a math problem is figuring out what the problem is asking you to do, so please share what you've figured out with your classmates!

Also, some students do not seem to be getting email at the address registered with the college. Could someone show them these emails or forward these to them? Thanks!

The question about the pentominoes: If my student ID# was 12345676, then the last three digits are 6,7, and 6. However, the last three distinct digits are 5,6, and 7. Therefore, the pentominoes I have to use in this problem are those three labeled with the numbers 5,6, and 7 in the picture on the exam. (And I have to choose two of them!) A more difficult problem for many of you seems to be figuring out what makes a tesselation or tiling with your chosen pentomino. Please help each other with this: the main three criteria:
(1) no gaps
(2) no overlaps
(3) the pattern clearly continues in all directions throughout the plane.

Thus that problem has nothing to do with the pentominoes with "names" x, y, and z!

I am not sure what the other question about the Hamiltonian/Euler graphs really is. (Hannah said there were highlighted blue and red sections, but I don't see those colors...) I'll try to do one more example and email or post that too.

Hannah also said there was another question, but did not say what it was. Someone let me know

Here's another street grid example.

Class 14, Thu., May 22, 2008
We worked on the take-home exams, and also started the chaper on fractals, working on the handout on fractal trees (complete and bring to class on Tuesday).
HW: Ch. 6.1 1,2,3,5,7,8
HW: Ch. 6.2 3,12-23

Here's some material on the game of life, which we learned about in class:
Executable Java applet that will run the game of life on the web; you can also download a free executable program: http://www.bitstorm.org/gameoflife/

Easy to understand background and explanation of the game of life:
http://www.math.com/students/wonders/life/life.html

We went over a variety of concepts related to chaos, fractals (see this link to fractal galleries), and "dynamical systems" (systems that change over time - here is a site with some neat animations). Here is another site devoted to fractals.
Here is a site with interactive software for creating fractals and learning about chaos.
Here's a site for looking at the Mandelbrot set.

Your next written report is due one week after the take-home test: Here is a list of books by Martin Gardner. Choose a chapter (or two) that is of interest to you and report on it:
The Colossal Book of Mathematics
Mathematical Circus
Mathematical Magic Show
The Magic Numbers of Dr. Matrix
Mathematical Carnival
Knotted Doughnuts and Other Mathematical Entertainments
Wheels, Life, and other Mathematical Amusements
Time Travel and Other Mathematical Bewilderments

Please check out one of the books and sign up to do a report on one chapter. You can email me which chapter you want to do a report on, if you have any question. This will be a 600 word paper, as before. The main criteria for how you select which chapter to report on is that it should be of interest to you.

Here is your f inal project form, which is due the last week of class. I will ask you to fill it out in the next week. Filling it out completely is part of your project grade (15%). I will also email everyone in the class list the form as an MS Word file, so you can more easily fill it out.

Class 13, Tue., May 20, 2008
We heard more mathematical biographies today. We also learned about Euler circuits and Hamiltonian cycles,
We saw how to "Eulerize" a graph by adding doubling back edges.
We also constructed a tetrahedron, octahedron, and cube with a loop of string. See this site for reminder illustrations on making string polyhedra.

Bring a length of thick string, about 5 feet long, to class on Thursday, and I'll teach you the knot trick!

By the way, here is a video story on some of our math/dance work.

Bring your take-home exams on Thursday, we'll work on them some in class.

We will start the next chapter on Thursday, so be up to date on homework by Thursday.

Class 12, Thu., May 15, 2008
Sorry this has taken so long, but here is your take-home exam.
We spent much of last class on math bios, and we'll finish on Tuesday.

Class 11, Tue., May 13, 2008
We spent the first hour working on exam corrections. Then we introduced some topology topics, including the torus, mobius strip, and knots.

I still need to take up the pentomino tiling, which I'll do on Thursday. Also, complete and bring the rotational symmetry handout I gave you today.

Your biographies are due Thursday. See below for more info.

HW, Ch. 5.2: Do any three of 8,9,10,13,14. Also do 18,19,22, any one of 26-28, 37
HW Ch. 5.4: 1-5,13,16,19,20

Class 10, Thu., May 8, 2008
We spent a lot of time going over the handouts on rotational symmetry and reflection symmetry. We also went over the graph theory alphabet, from section 5.3.

I'll take up that handout with the pentomino tiling on Tuesday.

Also on Tuesday your corrections on the exam are due. I've set up a class discussion site at
http://groups.google.com/group/math44spring08
You should have received an email explaining how to join the group. Please use the group to discuss the problems and corrections.

Someone asked me about the problem about pouring water. This problem is just like the problem explained on page 20 of the text.

Your bio paper are due next Thursday. Let me know if you are still in need of materials for your paper.

For now we'll skip sections 4.6 and 4.7. We'll go over section 5.2 on Tuesday.
HW, Ch. 5.1: 4-6, 12,14,16,22,25,26,36
HW: Ch. 5.3: 2,4,5,7-10

Class 9, Tue., May 6, 2008

We examined tilings or tesselations with pentominoes.
(1) Please finish your tiling with a pentomino and bring to class on Thursday. Try to find all the symmetries in the tiling, and indicate on it exactly where they are. Here's a handout showing you how to find the symmetries in your tiling! Here are hundreds of links to some amazing tiling sites.

(2) Bring the Poinsot Stars handout to turn in on Thursday.
(3) Also, complete the houndouts on rotational and reflection symmetry given out in class today.

(4) Important: Your corrections to exam 1 are due next Tuesday. In order to receive 1/2 the points missed, you must:
(a) Correct all incorrect answers on the exam itself near the incorrect answer - do not use extra sheets. (Write small!)
(b) Make a short statement explaining what your error was, if it was a problem that did not originally require an explanation.
(c) Cross out the incorrect answers with a single line (do not erase) so that I can see what the original error was.
(d) Use a different color ink or pencil for your corrections, and circle the correct answer.
(e) If all corrections are correct, you will receive 1/2 of missed points on the exam.
(f) Turn in at start of class on Tuesday.
(g) You may work together outside of class, but your work may not look exactly like someone elses!
(h) We won't take any more class time. Be prepared to stay after class, or work with other students before class, if you want to work together.

(5) Your math biography is due next Thursday, May 15, please let me know if you need references.
Here is the list showing who is assigned whom!
If you were not in class, you need to choose someone to report on; email me who you plan to report on by Thursday. The reference books, Mathematical People and More Mathematical People should be on reserve in the campus library. The report will be 600 words, and will also involve a short 2 to 3 minute oral report. Include in your paper some response by yourself to the person - is this someone you might like to have a conversation with, or is it someone who does not seem very interesting to you? Your main reference must be a printed source, not a web site, and you must cite any sources you use. Due Th. May 15.
John Conway- Here's an additional article
I also have an additional article on Dirk Struik, that I'll pass along to James, and Ken, you might want to look up the article a number of years ago, I think in Atlantic Monthly, on Paul Erdos.

Those of you who don't yet have someone assigned to you (Yuenman, Renecia, Angela), check out these:
Women Mathematicians - If you'd like to report on several contemporary mathematicians with bios here, let me know
Mathematicians of the African Diaspora - If you'd like to report on several contemporary mathematicians with bios here, let me know
Richard K. Guy - I will supply an interview article for Ginger.
Jean Taylor - I will supply an interview article
Let me know who you are interested in reporting on.

Please visit Scott Kim's homepage and look at the many examples of inversions.

Also looked at slides of polyhedra and related mathematical objects, including "DNA Origami". Here is Paul Rothemund's home page, you can find much more on this subject there.

We'll start chapter 5 on Thursday.

Class 8, Thu., May 1, 2008
We went over material from chapters 4.3 and 4.5, and some of ideas about symmetry related to section 4.4.

We also went over Poinsot Stars please turn in the handout on Tuesday.

Also, please hand in on Tuesday one tiling or tessellation with one pentomino, as we went over today.

I'll place more info on the next paper here shortly.

Finish the pentomino sequence problem, bring your answers on Tuesday. See the hint at the diagram of the 12 pentominoes that I suggested you use to solve the problem. Don't look if you are still working on it yourself!

You really should check out some of the amazing web sites about polyhedra, for example George Hart's Pavilion of Polyhedrality (you may need to update your software to see everything.) Hart seems to be channeling some kind of alien polyhedral knowledge.
The Math Forum site http://mathforum.org/alejandre/applet.polyhedra.html has a nifty Java Applet that allows you to choose and rotate various polyhedra.

HW Ch. 4.3 # 1-8
HW Ch. 4.4 # 1-5
HW, Ch. 4.5: 11,12,15,16,21

Class 7, Tue., April 29, 2008
We had the first exam. We'll continue with chapter 4 on Thursday. Bring the Poinsot Stars handout, we'll do some work on it in class.

Class 6, Th., April 24, 2008
We went over homework problems, also some mod arithmetic material. We also started on chapter 4.1 and 4.2.
We went over more of the pentomino problem. Please complete by Tuesday.
Bring homework to class on Tuesday, I'll check during your first exam.
Here is a list of sample exam questions.
HW: Ch. 4.1 #2,11,12,13,15,18,19
HW: Ch. 4.2 #5,7,9,11,21

Class 5, Tue., April 22, 2008
We went over a number of homework problems, and introduced sections 2.4 and 2.5.
On Thursday, bring to class: Poinsot Stars - we'll talk about this in class.

Here's a simple example of how the RSA code works.

Here's a list of "distributed" mathematical projects now being conducted throughout the internet.
Here's the "World Community Grid" site listing many online projects in which you (and your computer) can participate!
Here's another site for "Volunteer Computer Grids."

We went over a little more of the pentomino sequence problem, bring your answers on Tuesday. See the hint at the diagram of the 12 pentominoes that I suggested you use to solve the problem. Don't look if you are still working on it yourself!

Here again is the list of mathematicians for your biography paper, due May 6. Choose one who seems interesting to you. You'll be required to use and cite a published (non-web) source, though you can use the web to do preliminary research in order to find one of interest.

Ch. 2.4 #3,4,5,7,9,19,28,29,30,32
Read ch. 2.5 and 4.1 and start work on:
HW Ch. 2.5: 3,6,7,8,11,15

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.

Here's a preliminary study guide for exam 1:
Pattern problem
Check digit codes
Fibonacci numbers
Number theory, primes
Pythagorean theorem
Pigeonhole Principle
Art Gallery Theorem

Class 4, Thu., April 17, 2008
We went over some homework, and also explored the Fibonacci numbers. We also looked briefly at prime numbers.

Due on Tues., April 22: Fibonacci number assignment. This will count as a "quiz."
Print and bring to class on Tues: Poinsot Stars - we'll talk about this in class.

We went over a little of the pentomino sequence problem, bring your answers on Tuesday. See the hint at the diagram of the 12 pentominoes that I suggested you use to solve the problem. Don't look if you are still working on it yourself!

Here is the list of mathematicians for your biography paper, due May 6. Choose one who seems interesting to you. You'll be required to use and cite a published (non-web) source, though you can use the web to do preliminary research in order to find one of interest.

New homework (I will check homework on Tuesday):
Ch. 2.3: 1-3,6,7,13,14,19,20,22,37,39

Class 3, Tue., April 15, 2008
We introduced the Fibonacci numbers and the pigeonhole principle.

Here are some links to Fibonacci sites:
A great site about Fibonacci numbers.
Here's another Fibonacci site with lots of pictures and interactive applets.
Here's an interactive site that helps explain phyllotaxis.

We also tried to find all the pentominoes. Print out the pentomino sequence problem. You might want to start working on these problems (they'll be due later next week). You might want to cut out small squares for the pentomino chain problem, or cut out actual pentominoes for the second set of problems.

Work HW problems for sections 2.1 and 2.2:
Ch. 2.1 I: 4,6,9,10-12,14-18,21
Ch. 2.2: 1-7,16,18,21,24,26,

Here are more references to counterfeit coin problems:
http://www.cut-the-knot.org/blue/weight3.shtml
http://www.maa.org/mathland/mathtrek_2_16_98.html
http://www.cut-the-knot.com/blue/weight1.shtml

Here are some links for polyominoes and pentominoes. There's lots more!

Class 2, Th., April 10, 2008
We worked more on the Frogs on a Log problem, and noticed some new things: if there are N frogs on each side, then the minimum number of moves seems to be N(N+2) = (N+1)² - 1. Also, we noticed that it will take 2N+1 more moves than the previous problem in which there were 1 fewer frogs on each side. But we have not yet figured out why? See if you can come to some conclusions.

We also worked on more of the problems from the first chapter. Continue working on them for homework. For example, for the 3 light bulb problem, see if you can extend the problem, and use a similar technique if there are 4 light bulbs and 4 switches. For all the problems, think about using objects to help you reason about the problem. One of the other problems we worked on was the 3 cats and dogs problem (called the "missionaries and cannibals" problem in the text - this is the older, not much used anymore name for that problem.)

We also worked on the pattern handout. Complete and turn in Tuesday.

By the way, here are some of the vocabulary words we used during the past two classes. Try to use each one in a sentence, to make sure you understand them:
symmety: we'll define this more carefully later, but for now, think of it as a design in which parts are repeated in some fashion.
multiple: 12 is a "multiple" of 3 and of 4. 3 and 4 are "factors" of 12. Is 13 a multiple of 1? Is 0 a multiple of 13?
horizontal (row): parallel to the horizon. Often means we are thinking about right and left.
vertical (column): up and down
odds: 1,3,5,7,... Is -1 and odd number?
evens: 0,2,4,6,8, .... Is -6 and even number? Is 0 even?
alternate: a pattern in which two "sub-patterns" are each displayed in every other section of the pattern.
inductive thinking: arguing from specific cases to a general rule
deductive thinking: arguing from a general rule to a specific case

Due on Tuesday:
(1) Math autobiography
(2) Pattern handout
(3) Problems from chapter 1

Class 1, Tue., April 8, 2008
We played the pattern game, and worked on the "frogs on a log" problem.

Please print out the handout Patterns and Modular Arithmetic, work the problems, and bring to class on Thursday.

Here are some links about modular arithmetic, which we will examine in more depth soon:
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.

HW due next Tuesday: Read and work all the problems at the end of chapter 1, #1-15, pg 28-32, during the first week. I will not collect these problems on Thursday, since not everyone will have their text by then, but we will work on them!

Paper due next Tuesday: Mathematical Autobiography, description on page two of the Green Sheet

We also worked on the "Frog Crossing" problem.
Three round frogs (O's) and three crossed frogs (X's) are sitting on a log with seven spaces, and want to take each other's places. A frog can move one step to a vacant square, or jump over one neighbor to a vacant square. Show how it can be done in 15 moves.

Two frogs on a three space log can take each other's places in three moves:

Can you explain how? Four frogs on a five space log can change places in 8 moves, as we discovered in class.

Use what you discovered to decide in how few moves eight frogs can change places:

Can you generalize what you have discovered?