Math 44, Winter 2013, Home Page

Math 44
Spring 2015 Home Page
Green Sheet

Class 22, Th., June 18, 2015
Review
Here's a final exam sample problem list from a previous quarter. Here's another list of sample problems.
And here are some solutions to the problems in the lists above.

Some more projects due, also bring the cellular automata handouts, the fractal dimension handouts, and all other handouts not yet turned in!

Class 21, Tu., June 16, 2015
We heard projects and worked on cellular automata handout

Class 20, Th., June 11, 2015
Sub Kejian Shi went over material from ch. 6

Class 19, Tu., June 9, 2015
Ch. 6 on fractals, etc.

Class 18, Th., June 4, 2015
Ch. 6 on fractals, etc.

Class 17, Tu., June 2, 2015
Worked on Take-Home exams.

Class 16, Th., May 28, 2015
We worked on several older problems from exam 1 and also on symmetry and on Eulerizations of graphs (related to the take-home exam, which is due next Thursday.)

Class 15, Tu., May 26, 2015
We heard reports based on paper #3, and went over material from chapter 5 on topology and surfaces.

Class 14, Th., May 21, 2015
We went over chapter 4 homework and did more with graphs.

Class 13, Tue., May 19, 2015
We learned the graph theory alphabet of Nicholas Branca and found all 11 "simple" (no loops or multiple edges) graphs with 4 vertices.

Class 12, Th., May 14, 2015
We learned how to make polyhedra with our fingers; see these two short videos about this:
Handmade Math
Polyhedra at Your Fingertips

We also spent some time on redoing some of the problems from the first exam.
And we practiced making tessellations with tetrominoes and finding their symmetries (see this handout).

There is an enormous amount of recent mathematics and art on the subject of "tilings" or "tessellations." Take a look at David Eppstein's enormous collection of links on tilings. He calls his collection of such links the Geometry Junkyard; it's very entertaining, in a mathematical and arts sort of way! Here's his page on "Polyominoes and other Animals." Link on pentominoes.
Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. There are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane.

We have a number of upcoming deadlines:

Tuesday, May 19: Symmetry handouts due.
Geoboard paths
Rotational Symmetry
Reflection symmetry

Thursday, May 21: Chapter 4 homework due (sections 4.1 to 4.5 only.)

Tuesday, May 26: Paper 3 due on Turnitin.com; see description below.

Thursday, June 4: Take-home exam due. I'll be emailing this to you shortly.

Tuesday, June 16: Final class projects due.

For your third written report, due on Tuesday, May 26, you are to report on a mathematical subject connected to this course, that has caught your interest. If you saw it, you may do a report on the dance concert The Daughters of Hypatia: Circles of Mathematical Women, Th., Apr. 30, 2015 at the De Anza College Visual and Performaing Arts Center. You may NOT report on the Fibonacci numbers or the Golden Ratio! The report must be at least 600 words, and may cover the mathematics from one to several chapters of a book from the following list; other books or sources may also be used. You must use published material, not just web sites, unless you get prior permission from the instructor, and you MUST cite your sources at the end of your paper. A short oral report to the class will also be required.

You should include in what you write and talk about:

(1) why you chose this topic,
(2) what you learned, and
(3) what you think about the subject in question.
(4) What you might like to find out about the subject in the future.

I suggest that you read a chapter or two from one of Martin Gardner's books on mathematics, he covers a wide variety of topics. Here is a list of a few of the 70 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

You might also want to look at books by Ian Stewart, Keith Devlin, or Ivars Peterson.

This will be a 600 word paper, as before, submitted to Turnitin.com. The main criteria for how you select which chapter to report on is that it should be of interest to you. You will also give a brief 2-3 minute oral report on what you learned to the class, on Tuesday, May 26.

Class 10, Th., May 7, 2015
We built the Platonic solids out of plastic pieces and went over some material from chapter 4.

Edgar did the 4 rows of 5 trick - see this link for more about the trick!

Class 9, Tue., May 5, 2015
We reviewed material for exam 1, which is this Thursday, when chapter 2 homework will also be due.

Here is a study guide for the first exam from a previous quarter.
Here is another practice exam from a previous quarter, more like what you will have, since it's a multiple choice/Scantron exam. Please bring the half page long scantron for your exam 1.

Class 8, Thu., Apr. 30, 2015
The first exam is next Thursday, May 7. Chapter 2 homework will be due at the end of the exam, and we will discuss chapter 2 homework on Tuesday, May 5. You should also be working on chapter 4 homework, since we have gone over several sections from chapter 4.

We learned about four symmetries:
translation: p-->p
reflection: p-->q
180 degree rotation: p-->d
glide reflection: p-->b

We also made a short symmetry dance and worked on some handouts. Please try to complete the one we worked on and bring to class. Here are the handouts on symmetry
Geoboard paths
Rotational Symmetry
Reflection symmetry

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.
Here is the wikipedia article on UPC codes, with lots of detail.

Link on pentominoes.
Here's a handout showing you how to find the symmetries in a tiling! Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. For example, we might try to solve the same problem for the "hexomino:" there are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane. These are not shown in complete detail; if you create a pentomino tiling it should extend further than these, and indicate how the pattern you found really could extend in all directions throughout the plane.

Class 7, Tue., Apr. 28, 2015
We heard the math biographies and started chapter 4, and also learned about several symmetries.
Chapter 2 homework will be due next Tuesday. We will have class on Thursday, some of you may wish to attend the performance on Thursday at 1:30 PM in the De Anza Visual and Performing Arts Center Theatre. You may use that as the subject of your next paper.

Details:
On Thursday at 1:30 PM the dance company I co-direct is doing a performance of The Daughters of Hypatia: Circles of Mathematical Women, which celebrates mathematical women throughout history. See more info here.

Class 6, Th, Apr. 13, 2015
We went over more material from chapter 2 on prime numbers and modular arithmetic. We started working on
Patterns and Modular Arithmetic.
Modular Arithmetic Intro handout.
We'll complete these soon, partly in class, so continue working on them outside of class!

Class 5, Tue., Apr. 21, 2015
We learned about polyrhythms and their relation to modular arithmetic, which is the subject of section 2.4.

Here is the list showing who's reporting on which mathematicians. (The links are now fixed!)

Please turn in your Fibonacci assignment, this Thursday - read the assignment to see what you need to include.

Class 4, Thu., Apr. 14, 2015
We worked on material from chapter 2 and also on some of the homework problems in chapter 1. For example, we saw a way to translate the officers and comedians problem into a geometric network problem in order to better see how to solve it. We also learned about the prime numbers and infinitude of the primes, as well as the Twin Prime Conjecture (the conjecture that there are an infinite number of twin primes, like 17 and 19) and the Goldbach Conjecture (the conjecture that every even number larger than 2 is the sum of two prime numbers).

Here's the Fibonacci assignment, which will be due this Thursday, April 23.
Your Math autobiography is due next Tuesday.

Here's an article about Yitang Zhang's work on the twin prime conjecture.

Class 3, Tue., Apr. 14, 2015
We did the Fido puzzle, learned a little more about the Pigeonhole Principle, were introduced to the Fibonacci numbers, saw Vi Hart's Fibonacci number video, and decided who's reporting on which mathematicians.

Here's the Fibonacci assignment, which will be due a week from Thursday, April 23.
Here are some more links on Fibonacci numbers:
Vi Hart's Fibonacci number videos.
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.

Here's a solution to the 12 coin problem from chapter 1 written up by John Conway - in case you want to see a solution!

Class 2, Thu., Apr. 9, 2015
I've uploaded everyone's name and email address to turnitin.com for your first paper.
ID for the class on Turnitin:
Class name: Math 44, Spring 2015
Password: MathRocksS15
Class ID: 9824929

We went over some of the homework problems, used the counters to work on the "cops and comedians" river crossing problem as well as the "frogs on a log" problem, and saw a mathematical magic trick that uses parity (odds and evens). We also briefly learned about the pigeonhole principle and saw another "magic trick" based on it.

When we played the "take-away" game we actually used ideas from modular arithmetic.

We went over a number of problems from chapter 1, including the water pouring problem turned into a" billiards" problem. Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.

All of these puzzles in chapter 1 are famous ones in what is known as "recreational mathematics." The water measuring problem, according to puzzle history expert David Singmaster and others, goes back to a book called "Annales Stadenses" compiled by the monk Abbot Albert in the year 1240 CE in Germany. The billiard ball solution method is apparently due to M.C.K. Tweedie in 1939. If given 3 containers holding x, y, and z gallons of water, there is no known simple method for determining which amounts can be measured if z happens to be smaller than x + y. For example, if you have three containers holding 6, 10, and 15 gallons, and an unlimited supply of water, which amounts 1 through 14 is it possible to measure exactly?

Here's a reference on the weighing problems. (Problem 15 in the chapter 1 homework is quite difficult.) Here's a link to the New York Times puzzle column (called Numberplay). Unfortunately, they only allow you to read it a few times before charging for access!

Class 1, Tue., Apr. 7, 2015
We will play the pattern game.
We will playthe take-away game, in which each player removes 1 or 2 counters on each move, the last player to move winning. We will develop a winning strategy involving leaving your opponent with a multiple of 3 on each turn, if possible.
Work on chapter 1 problems 1-15, due at the fourth class, Thursday of next week.
Your math autobiography is due next Tuesday (see the green sheet above.)
Please get your textbook right away!

Papers for this class:
(1) Math Autobiography (see second page of green sheet). Due third class session.
(2) Math biography assignment: links to interviews and articles
The reference books, Mathematical People and More Mathematical People are 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.
(3) For your third written report, due on Tuesday, May 12, you are to report on a mathematical subject connected to this course, that has caught your interest. You may do a report on the dance concert The Daughters of Hypatia: Circles of Mathematical Women, Th., Apr. 30, 2015 at the De Anza College Visual and Performaing Arts Center. You may not report on the Fibonacci numbers or the Golden Ratio! The report must be at least 600 words, and will cover the mathematics from one to several chapters of a book from the following list; other books or sources may also be used. You must use published material, not just web sites, unless you get prior permission from the instructor, and you MUST cite your sources. A short oral report to the class will also be required.

You should include in what you write and talk about:

(1) why you chose this topic,
(2) what you learned, and
(3) what you think about the subject in question.
(4) What you might like to find out about the subject in the future.

(4) Project description

Study guides for first and final exam:
Here is a study guide for the first exam from a previous quarter.
Here is another practice exam from a previous quarter, more like what you will have, since it's a multiple choice/Scantron exam. Please bring the half page long scantron for your exam 1.

Here's a final exam sample problem list from a previous quarter. Here's another list of sample problems.
And here are some solutions to the problems in the lists above.

Some links from previous quarters:

Here is a short handout on 1-dimensional cellular automata, like we did in class.

Here is the site with the 4 rows of 5 trick that I did in class.

Film about Martin Gardner and learned about Euler Circuits.

We also looked at the "Small World Phenomenon," and "six degrees of separation,"
Here are some more links:
Six degrees of Kevin Bacon.
Oracle of Bacon, where you can look up actor's links to other actors.
Kevin Bacon's charitable SixDegrees.org.
The Erdos-Bacon number site.
The Facebook Six Degrees site.
The Erdos Number project.

Handouts on symmetry
Geoboard paths
Rotational Symmetry
Reflection symmetry

The "billiard method" of solving water pouring problems.
Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.

Sophie Germain's story. Here's a nicely written bio about Germain.

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

Here is a short handout on modular arithmetic; please print and birng to class Tuesday.

Here is the wikipedia article on UPC codes, with lots of detail. The reason it did not "work" in class the other day: we were looking at the bar code for the ISBN, not the UPC!

We also made rhythms out of our names, and saw how to put two rhythms together to make a star polygon. Here's a solution to the 12 coin problem written up by John Conway.

Here is the Fibonacci assignment,

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 1,2, and 3 are true. For Thursday, can you 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!

Here is the Fibonacci assignment, which is due next Thursday, Jan. 24.

Here are Vi Hart's Fibonacci number videos.
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.

Here is an article published in the New York Times about Martin Gardner, in honor of his 95th birthday.

Symmetry leads us to the field of "tilings" or "tessellations." There is an enormous amount of recent mathematics and art on this subject. Take a look at David Eppstein's enormous collection of links on tilings. He calls his collection of such links the Geometry Junkyard; it's very entertaining, in a mathematical and arts sort of way!

Link on pentominos.
Here's a handout showing you how to find the symmetries in a tiling! Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. For example, we might try to solve the same problem for the "hexomino:" there are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane. These are not shown in complete detail; your penomino tiling should extend further than these, and indicate how the pattern you found really could extend in all directions throughout the plane.

Rotational symmetry handout.
4-Fold rotational symmetry handout.

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."

Patterns and Modular Arithmetic.
Modular Arithmetic Intro handout.

Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.

We also played the "take-away" game, and saw an analysis of it that introduced modular arithmetic. And we learned a little about the history of modular arithmetic.

Visit Scott Kim's homepage and look at the many examples of Inversions, some from his book by the same name, some animated. Recently other artists have produced these letterform images that exhibit symmetry, for example here is a site for ambigram tattoos by Mark Palmer. The artist John Langdon has also been creating Ambigrams for many years, and produced those used by Dan Brown in his book Angels and Demons. See Langdon's site for examples of his work, or look for his book Wordplay. Here's a site by the mathematician and artist Burkard Polster with lots of playful examples.

Symmetry leads us to the field of "tilings" or "tessellations." There is an enormous amount of recent mathematics and art on this subject. Take a look at David Eppstein's enormous collection of links on tilings. He calls his collection of such links the Geometry Junkyard; it's very entertaining, in a mathematical and arts sort of way!

We also played the "take-away" game, and saw an analysis of it that introduced modular arithmetic. And we learned a little about the history of modular arithmetic.

All of these puzzles are famous ones in what is known as "recreational mathematics." The water measuring problem, according to puzzle history expert David Singmaster and others, goes back to a book called "Annales Stadenses" compiled by the monk Abbot Albert in the year 1240 CE in Germany. The billiard ball solution method is apparently due to M.C.K. Tweedie in 1939. If given 3 containers holding x, y, and z gallons of water, there is no known simple method for determining which amounts can be measured if z happens to be smaller than x + y. For example, if you have three containers holding 6, 10, and 15 gallons, and an unlimited supply of water, which amounts 1 through 14 is it possible to measure exactly?

Here's a reference on the weighing problems. Here's a link to the New York Times puzzle column (called Numberplay). Unfortunately, they only allow you to read it a few times before charging for access!

Here is the Fibonacci assignment, which will be due one week from Tuesday on Tuesday, Oct. 7. Read it CAREFULLY and do everything it asks!!

Here are some more links on Fibonacci numbers:
Vi Hart's Fibonacci number videos.
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.

Here is a short handout on 1-dimensional cellular automata, like we did in class.
We had project presentations, went over take-home exam.
Here's a sample problem list from a previous quarter. Yours will be similar, but there are some things here we did not go over, or did not go over very much. Here's another list of sample problems. Again, some similarities, some differences.

Here is the site with the 4 rows of 5 trick that I did in class.
Chapter 5 homework is due next Thursday, the last chapter homework is due on the day of the final exam.

How to tie a knot without letting go of the ends of a piece of string, and how to make a tetrahedron, five pointed star, octahedron, cube, pentagram, and six-pointed star with a loop of rope. Here are some links to videos showing how to make some of these figures.

Please print Patterns and Modular Arithmetic, work on the problems, and bring to next class.

Here is a site where you can communicate about study groups for the take-home, etc.

We will spend more time on polyhedra and symmetry next class. Here are two videos on the finger polyhedra:
Scott Kim talking about and demonstrating the four-finger tetrahdron.
The four-hand tetrahedron, and others.

Here is a set of puzzles created for the 100th birthday of Martin Gardner by the Grabarchuk family.

Link on pentominoes.
Here's a handout showing you how to find the symmetries in a tiling! Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. For example, we might try to solve the same problem for the "hexomino:" there are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane. These are not shown in complete detail; your pentomino tiling should extend further than these, and indicate how the pattern you found really could extend in all directions throughout the plane.

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 an article about Yitang Zhang's work on the twin prime conjecture.

Here's a list of "distributed" mathematical projects now being conducted throughout the internet, which we discussed last class. 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."

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.
Here is the wikipedia article on UPC codes, with lots of detail.

We also looked at prime numbers. Here is a 2 minute slightly nutty but mathematically accurate music video (professionally produced by Nova) of SJSU's Dan Goldston's discoveries about the Twin Prime Conjecture.

See modular arithmetic. And we learned a little about the history of modular arithmetic. Here are two further handouts:
Patterns and Modular Arithmetic.
Modular Arithmetic Intro handout.

Fibonacci assignment.