Fall 2016 Math 46 Home Page

Green Sheet
Textbook homework list

Class 18, Thu., Mar. 3
We went over material from chapter 7 on terminating, non-terminating, and repeating decimals. We discovered that if 1/n is a terminating decimal then n is the product of powers of 2 and powers of 5. We also went over some questions about decimals and percentages.

Next paper is due on Turnitin on Tuesday of next week, Mar. 8.
Homework for chapter 6 is due next Thursday, Mar. 10
Thursday of next week we'll do the Barbie activity.

Second Essay Assignment.
You have a short paper on a subject related to the course that catches your interest due Tue, Dec. 2 (two weeks later than originally assigned in green sheet), and worth 5% of your grade. You will turn the paper in via Turnitin.com. If you reported on the math/dance concert last time, you may do your math autobiography this time.

Here's the description of the essay:
Report on an article or chapter from a popular book about mathematics or math education. The report will be one to two pages long, typewritten, (it 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 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.

Examples of books with mathematical content:
The Mathematical Tourist and Islands of Truth, by Ivars Peterson.
Any of the books of Martin Gardner on mathematics (over 15 titles).
Game, Set, Math and Does God Play Dice by Ian Stewart, or other titles on math by Stewart.
The Mathematical Experience by Davis and Hersh.
A Number For Your Thoughts and Numbers At Work and At Play by Stephen P. Richards.
Tilings and Patterns by Grunbaum and Shepard.
Mathematical Snapshots by Steinhaus.
Mathematics: The New Golden Age by Keith Devlin, or other titles by Devlin.
The Emperor's New Mind by Roger Penrose.
The Mathematics of Games by John Beasley.
Archimedes' Revenge by Paul Hoffman
What is Happening in the Mathematical Sciences, ed. by Barry Cipra, Vols 1-5 (on reserve in campus library)

Examples of books with cultural content:
Ethnomathematics by Marcia Ascher.
You can also consult this Multicultural Mathematics Bibliography. Many of the references are in our library, and the bibliography contains call numbers for those that are in the library.
A number of Martin Gardner's books are in the De Anza library.

Class 17, Tue, Mar. 1
We learned more about fractions, as well as Egyptian unit fractions.

Class 16, Thu, Feb. 25
We went over material from chapter 6 on fractions and played Frac Jack.
I moved your due date for the take-home exam back to Thursday, Mar. 3.

Class 15, Tue, Feb. 23
We went over material on fractions from chapter 6.

Class 14,Thu., Feb. 18
We went over some chapter 4 problems, also started on chapter 6 on fractions and ratios, and worked on the take-home exams.
Please turn in chapter 5 homework next Thursday, Feb. 25.

Here's the 4 rows of 5 trick.

Class 13,Tue., Feb. 16
We went over chapter 5 material on positive and negative numbers, and learned different ways of adding and subtracting positives and negatives, using manipulatives (counting method) and a "walking" number line (measurement method). We also worked a little on the take-home exams.

Class 12,Thu., Feb. 11
We worked on group problems, and almost finished! I'll be emailing you your take-home problems. Print them and bring to every class for the next two weeks! We'll also start chapter 5 on Tuesday. Please turn in chapter 4 homework on Thursday, Feb. 18, of next week.

Class 11, Tue., Feb. 9
We went over material from chapter 4 and also worked on the group problems, which will be presented on Thursday.

Please print out and bring to class this handout with problems on Patterns and Modular Arithmetic
The Hidden Role of Modular Arithmetic, that reviews what we did in the first two classes and relates it to some other problems from chapter 1.
Here are some links about modular arithmetic, which we will learn more about throughout the quarter:
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.

Class 10, Thu., Feb. 4
We went over material from chapter 4 on greatest common divisor (gcd) and least common multiple (lcm).
We also worked on group problems. We will meet in problem groups on Tuesday to finalize solutions to problems.

Class 9, Tue., Feb. 2
We went over material from chapter 4.1 and 4.2.

Below are the group problem assignments. Chapter 3 homework is due this Thursday.

Article by Brian Hayes on the history of Gauss's Trick, published in 2006.
Here are articles on Nines Complement subtraction and the Gelosia mulitplication method.
Here are some other methods of addition, etc.

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.

Class 8, Thu., Jan. 28
We began chapter 4 with divisibility methods, and I showed you the remainder when dividing by 9 trick.
Homework for chapter 3 will be due next Thursday, Feb. 4.

We had exam 1.

Class 7, Tue., Jan. 26
We went over material from chapter 3 on mental arithmetic and estimation methods.
Homework for chapter 2 will be due Thursday, Jan. 28.

We did the class activity on mental arithmetic versus paper and pencil versus calculator.
We went over alternative calculation methods, including Gelosia or lattice multiplication, subtraction

Class 6, Th., Jan. 21
We went over material from chapters 3.1 and 3.2.
At the end of class I demonstrated Nines Complement subtraction.

Study guide for the first exam
Solutions to the study guide for the first exam problems
Here's another old exam 1.

Class 5, Tue., Jan. 19
We went over questions from chapter 1 and 2 and went over new material from 2.3 and 2.4.
Chapter 1 homework is due on Thursday of this week.
Some of you have been asking: the only things to turn in on Turnitin.com are the two 600 word papers, NOT the homework, which should be turned in on paper.

Class 4, Thu., Jan. 14
The subsitute Kejian Shi went over material from chapter 2.1 and 2.2. You should now be working on chapter 2 homework.

Class 3, Tue., Jan. 12
The substitute Kejian Shi went over material from chapter 1. Please be working on all sections of chapter 1 homework, listed at the link at the top of this page. Chapter one homework is due a week from Thursday, on Thur. Jan. 21.

Here is the
Fibonacci assignment which will be due Tuesday, Jan. 19.

Please be working on your journal!

Class 2, Thu., Jan. 7, 2016
Your mathemtatical autobiography is due on Tuesday on Turnitin.com.

We learned about several sequences of numbers, including:
The odds: 1,3,5,7,...
The evens: 0,2,4,6,...
The squares: 1,4,9,16,...
The triangular numbers: 1,3,6,10,15,...
The cubes: 1,8, 27, 64, 125, 216,...
The Fibonacci numbers: 1,1,2,3,5,8,13,21,24,...

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 a difference of 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 1-3 are true. Number 5 is tricky, and involves labeling the six pigeonholes with different numbers of numbers!

Here is the
Fibonacci assignment which will be due Tuesday, Jan. 19.

You should be working on homework from chapter 1. I'll try to upload scans of the homework, though not of the text from chapter 1.

Class 1, Tue., Jan. 5, 2016
Homework:
(1) Get a looseleaf notebook and set up sections as described on the third page of the green sheet or below.
(2) Establish a journal etherpad site. Email me the URL of your site. (See directions below or on the green sheet.)
(3) Read and work on homework from sections 1.1-1.2. Chapter 1 homework will be due on Thursday, Jan. 15. The list of homework problems is below.
(4) I have registered you to the Turnitin.com class page with the email in my class record, so make sure you can access the class site at which you will be uploading your essays. Your first essay, your mathematical autobiography, is due on Tuesday, Jan. 13, by 4 PM. It is described above in the green sheet. Read the description of the assignment as you write it to make sure you include what it asks for!

Here is the pattern game we play in class. Print this out and include in your portfolio.

We will play the take-away game, in which each player removes 1 or 2 counters on each move, the last player to move winning.
We saw that the losing postions are all multiples of 3. If a player may remove 1, 2, or 3 counters on each move, the losing postions are all multiples of 4.

Here are some of the vocabulary words we have used during class. Try to use each one in a sentence, to make sure you understand them:
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
odd numbers : 1,3,5,7,... These numbers are "congruent" to 1, mod 2. Is -1 an odd number? (We'll learn about number "congruence" soon!)
even numbers : 0,2,4,6,8, .... Is 0 even? These numbers are "congruent" to 0, mod 2. Is -6 and even number?
alternate: a pattern in which two "sub-patterns" are each displayed in every other section of the pattern.

Portfolio. Put together your portfolio, a loose leaf notebook with these sections:
Homework
Handouts or articles provided to you at this site (for example the pattern game handout.)
Exams
Class notes
Articles
Your papers or essays

Write a journal entry for each week of class. It should be one long (6 or more sentences) several short paragraphs detailing your reflections on each day’s class. What struck you as interesting, useful, helpful, unhelpful, puzzling, etc.? How are you feeling about the class? What are your expectations of the class and your own participation? Imagine you are writing to your future self (as in a popular South Park episode?!) and mention those things most memorable! Keep your journal entries at a page you get at an etherpad site, for example, at the Mozilla etherpad site (at Mozilla click on "Create new public pad.")

Use this format for journal entries:
Stanley Student (keep your name at the top)

Th. Jan. 12 (most recent entry)
Blah, blah, blah (at least 1 long - 6 or more sentences - or 3 medium size paragraphs).

Tue. Jan. 10 (older entry)
Blah, blah, blah (at least 1 long or 3 medium size paragraphs).

If you have trouble using an etherpad site, try opening it with a different browser. I have no trouble using the (free) Google Chrome browser.

Links from previous classes:

Study guide for the first exam
Solutions to the study guide for the first exam problems
Here's another old exam 1.
Math practices from the Common Core

Recent final exam study guide
Note that the study guide is from a previous year; you won't have an essay question.
Recent sample problem sheet.
Sample problems with some hints and solutions.

Handout on Egyptian Fractions,

Slides of fraction problems.

4 rows of 5 trick

TED Dan Meyer video.

Group Problems:

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.
Patterns and Modular Arithmetic.
Modular Arithmetic Intro handout.

Second Essay Assignment.
You have a short paper on a subject related to the course that catches your interest due Tue, Dec. 2 (two weeks later than originally assigned in green sheet), and worth 5% of your grade. You will turn the paper in via Turnitin.com. If you reported on the math/dance concert last time, you may do your math autobiography this time.

Here's the description of the essay:
Report on an article or chapter from a popular book about mathematics or math education. The report will be one to two pages long, typewritten, (it 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 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.

Examples of books with mathematical content:
The Mathematical Tourist and Islands of Truth, by Ivars Peterson.
Any of the books of Martin Gardner on mathematics (over 15 titles).
Game, Set, Math and Does God Play Dice by Ian Stewart, or other titles on math by Stewart.
The Mathematical Experience by Davis and Hersh.
A Number For Your Thoughts and Numbers At Work and At Play by Stephen P. Richards.
Tilings and Patterns by Grunbaum and Shepard.
Mathematical Snapshots by Steinhaus.
Mathematics: The New Golden Age by Keith Devlin, or other titles by Devlin.
The Emperor's New Mind by Roger Penrose.
The Mathematics of Games by John Beasley.
Archimedes' Revenge by Paul Hoffman
What is Happening in the Mathematical Sciences, ed. by Barry Cipra, Vols 1-5 (on reserve in campus library)

Examples of books with cultural content:
Ethnomathematics by Marcia Ascher.
You can also consult this Multicultural Mathematics Bibliography. Many of the references are in our library, and the bibliography contains call numbers for those that are in the library.
A number of Martin Gardner's books are in the De Anza library.

Nines Complement subtraction and the Gelosia mulitplication method.
Some other methods of addition, etc.

Here's a site on Base systems.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.
Site on number systems.
How things work on basketball players' numbers: "Each uniform must display one or two digits on the front and back of the jersey. The numbers on a jersey are used to identify a player when calling violations. In most cases, the digits can only be 0, 1, 2, 3, 4 or 5. While the NBA has allowed players to use numerals higher than 5, it is a rare allowance. This limitation on numerals allows referees to use their hands to signal player numbers to the game's official scorekeeper. Otherwise, a player wearing number 9 could be confused with a player wearing number 54."

The "billiard ball method" of solving water pouring problems.The water measuring problem appears in one of the Die Hard movies, and 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 - the billiard method is not that helpful in this case. 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 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.
Here's an article explaining the method.

Handout about the Fido puzzle from class: Where's Fido?

Fibonacci assignment.
Sorting Junk game

New Math and how it grew out of the launch of Sputnik - see this article.
The symbolism of set theory and logic was developed during the late 1800s and later to try to make all mathematics into an algebraic activity, and how that failed totally due to the work of Kurt Godel, whose Incompleteness Theorem is based on the self-referential "Liars Paradox."

Inductive versus deductive thinking.

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 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, 3, and 4 are true. For Thursday, can you use the pigeonhole principle to explain property 5? 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!

Pascal's triangle, why the sum of successive odd numbers produces a square number, and a visual proof of the Pythagorean Theorem.

Here are Vi Hart's Fibonacci number videos.
Here is a great site about Fibonacci numbers.
Here is a Fibonacci site with lots of pictures and interactive applets.
Here is an interactive site that helps explain phyllotaxis, which is the pattern of spirals in many plants.
Here is a site about Pingala's possible use of the Fibonacci Numbers in ancient India.
Look up Rachel Hall who credits Indian mathematician/musician with the Fibonacci numbers - see her article "Math for Poets and Drummers" listed at her site.

Ken Ken site
Here's a handout with a Ken Ken example, with solution explained, which may help you understand how the puzzle works.

We learned about several number sequences which are important, including
The odds: 1,3,5,7,...
The evens: 0,2,4,6,...
The squares: 1,4,9,16,...
The triangular numbers: 1,3,6,10,15,...
The cubes: 1,8, 27, 64, 125, 216,...
The primes: 2,3,5,7,11,13,...
The Fibonacci numbers: 1,1,2,3,5,8,13,21,24,...

Here's the "Clap your name" activity.
Here are the
Common Core Standards,
National Council of Teachers of Mathematics

We played the 15-sum game, and how it is really "Magic Square tic-tac-toe."
We played the game Ken Ken. Please go to the Ken Ken site and practice playing the game!
Here's a handout with a Ken Ken example, with solution explained, which may help you understand how the puzzle works.

Here is the set of slides of fraction problems.
Here is the set of slides of decimal/ratio problems.
Farey Sequences (skip the advanced part and the cute animation at the beginning!)

Patterns and Modular Arithmetic.

Ken Ken Example

Here are articles on Nines Complement subtraction and the Gelosia mulitplication method.
Here are some other methods of addition, etc. Here's another such web site.

Here's a site on Base systems.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.
Site on number systems.

We did the Fido puzzle - please print out this Where's Fido handout and include in your portfolio. Include the extra problems in the handout as part of homework!

Here is the set of slides of fraction problems.
Here's a related handout on star polygons. Here are some materials on "modular arithmetic," which explains some of the properties we observed when looking at the rhythm patterns within circles:
Handout with problems on Patterns and Modular Arithmetic
The Hidden Role of Modular Arithmetic, that reviews what we did in the first two classes and relates it to some other problems from chapter 1.
Here are some links about modular arithmetic, which we will learn more about throughout the quarter:
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.

By the way, how is Halloween like Christmas?
Because Oct 31 = Dec 25 (do you know what this refers to? Hint: base systems...)
Here's a handout with a Ken Ken example, with solution explained, which will help you with one of the take home problems.

Here are articles on Nines Complement subtraction and the Gelosia mulitplication method.
Here are some other methods of addition, etc. Here's another such web site.

Here's a site on Base systems.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.
Site on number systems.

How things work on basketball players' numbers: "Each uniform must display one or two digits on the front and back of the jersey. The numbers on a jersey are used to identify a player when calling violations. In most cases, the digits can only be 0, 1, 2, 3, 4 or 5. While the NBA has allowed players to use numerals higher than 5, it is a rare allowance. This limitation on numerals allows referees to use their hands to signal player numbers to the game's official scorekeeper. Otherwise, a player wearing number 9 could be confused with a player wearing number 54."

Here is the Fibonacci assignment, which is due next Tuesday, Feb. 4. Read it CAREFULLY and do everything it asks!!

If you have not gotten an etherpad site for your class journal, do so now.
If you have not entered a journal entry for Tuesday also, please do so now, as I am already checking journals. Your first journal check will be complete by Tuesday, April 24, and will count 2 points of the 10 points for the "Portfolio" portion of your grade.

Here are some of the vocabulary words we have used during classes. Try to use each one in a sentence, to make sure you understand them:
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
odd numbers : 1,3,5,7,... These numbers are congruent to 1, mod 2. Is -1 and odd number?
even numbers : 0,2,4,6,8, .... Is 0 even? These numbers are congruent to 0, mod 2. Is -6 and even number?
alternate: a pattern in which two "sub-patterns" are each displayed in every other section of the pattern.

We also learned about a connection between the patterns in this game and the 3 by 3 magic square. Here's a handout on magic squares and modular arithmetic. We haven't covered all of the material in this handout yet, but will soon.

We will see several short videos about learning and teaching; you can find the links within Keith Devlin's recent online column.
Here is the set of slides of fraction problems.
Here is the set of slides of decimal/ratio problems

"PEMDAS" memory mnemonic

Prediction Card Trick handout
Painting the Pool
Britney Gallivan, who folded a "sheet" of paper 12 times.
Here is the set of slides of fraction problems.
Here is the set of slides of decimal/ratio problems.
Farey Sequences (skip the advanced part and the cute animation at the beginning!)
TED Dan Meyer video.

Nines Complement subtraction.
Gelosia mulitplication method
"Clap your name" activity.
Wikipedia entry on Turnitin.
Common Core Standards,
National Council of Teachers of Mathematics.

The triangle numbers,
Base systems
How things work on basketball players' numbers: "Each uniform must display one or two digits on the front and back of the jersey. The numbers on a jersey are used to identify a player when calling violations. In most cases, the digits can only be 0, 1, 2, 3, 4 or 5. While the NBA has allowed players to use numerals higher than 5, it is a rare allowance. This limitation on numerals allows referees to use their hands to signal player numbers to the game's official scorekeeper. Otherwise, a player wearing number 9 could be confused with a player wearing number 54."
Keith Devlin's articles on multiplication as repeated addition.
Brief history of the New Math.
The game Set, see their daily puzzle.
Sorting Junk game.

Here's a quote from Lewis Carroll's Through the Looking Glass. Alice is talking with the White Knight, who many commentators believe to be a stand-in for Carroll himself. We'll see it's relevance later in the course!
"The name of the song is called 'Haddock's Eyes'."
"Oh, that's the name of the song, is it?" Alice said, trying to feel interested.
"No, you don't understand," the Knight said, looking a little vexed. "That's what the name is called. The name really is 'The Aged Aged Man'."
"Then I ought to have said 'That's what the song is called?'" Alice corrected herself.
"No, you oughtn't: that's quite another thing! The song is called 'Ways and Means': but that's only what it's called, you know!"
"Well, what is the song, then?" said Alice, who was by this time completely bewildered.
"I was coming to that," the Knight said. "The song really is 'A-sitting on a Gate': and the tune's my own invention."

Voting methods and their history.
Where's Fido?

How things work on basketball players' numbers: "Each uniform must display one or two digits on the front and back of the jersey. The numbers on a jersey are used to identify a player when calling violations. In most cases, the digits can only be 0, 1, 2, 3, 4 or 5. While the NBA has allowed players to use numerals higher than 5, it is a rare allowance. This limitation on numerals allows referees to use their hands to signal player numbers to the game's official scorekeeper. Otherwise, a player wearing number 9 could be confused with a player wearing number 54."

Al Khwarizmi.

History of the Magic Square.

Triangle numbers, squares, Fibonacci numbers.

The game Ken Ken - this site has 6 new puzzles every day.

Article by Brian Hayes on the history of Gauss's Trick, published in 2006.
Here are articles on Nines Complement subtraction and the Gelosia mulitplication method.
Here are some other methods of addition, etc.

The pattern game we played in class
Handout with problems on Patterns and Modular Arithmetic
The Hidden Role of Modular Arithmetic, that reviews what we did in the first two classes and relates it to some other problems from chapter 1.
Here are some links about modular arithmetic, which we will learn more about throughout the quarter:
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.

The 15-sum game, and how it is really "Magic Square tic-tac-toe."

Here's a handout about the Fido puzzle from class: Where's Fido?

The game Set can be found here as a daily puzzle.
Here is a handout on what I call the Sorting Junk game.

The "Frogs on a log" problem, which is a textbook homework problem, and in which we will find that with
1 frog per side it took a minimum of 3 moves to exchange places
2 frogs per side it took a minimum of 8 moves to exchange places
3 frogs per side it took a minimum of 15 moves to exchange places
You might have guessed that 4 frogs per side would require 24 moves, 5 frogs per side would require 35 moves.
Figure out how to do these exchanges also, in the minimum number of moves.

We went over some homework problems from chapter 1, focusing attention on some number sequences that we'll be seeing a lot of this quarter:
The odds: 1,3,5,7,...
The evens: 0,2,4,6,...
The squares: 1,4,9,16,...
The triangular numbers: 1,3,6,10,15,... (see Triangle numbers for more on these.)
The cubes: 1,8, 27, 64, 125, 216,... (whooops, forgot to mention this one!)
The primes: 2,3,5,7,11,13,...
The Fibonacci numbers: 1,1,2,3,5,8,13,21,24,...
By the way, there is something called the Online Encyclopedia of Integer Sequences (OEIS), started by mathematician Neil Sloane, at which you can find just about any number sequence you can imagine!
We also looked at the Pascal (also called the Pascal-Khayyim) triangle.
We saw visualizations of some of these sequences, and noted that visual mathematics has become more popular recently as technology has supported more visual displays, and this has even led to changes in language taking us back in the direction of ancient hieroglyphics! (Think emojis or emoticons.) P

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 a difference of 1.
(5) A pair of your numbers had the property that one divided the other equally.
(6) 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 more difficult to explain; in this case the we have to show that there are more possible subsets than there are possible sums.

"Frogs on a log" problem, which is a textbook homework problem in section 1.6, and in which we will find that with
1 frog per side it took a minimum of 3 moves to exchange places
2 frogs per side it took a minimum of 8 moves to exchange places
3 frogs per side it took a minimum of 15 moves to exchange places
You might have guessed that 4 frogs per side would require 24 moves, 5 frogs per side would require 35 moves.
Figure out how to do these exchanges also, in the minimum number of moves.