Please spend at least 1 hour EVERY night reading the material/looking at the proofs/making sure you understand the details. Below is a reading list and homework assignments from when I taught this as an independent study. It is subject to changes depending on the amount of material covered each week, who ends up taking the course, and a variety of other factors (including the fact that we'll have a textbook). I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day. The items below are meant to be a guide.

• Week 13: Nov 24, 2014:

  • Return of Project Euler: https://projecteuler.net/

    • Make sure you have a Project Euler account.

    • Make sure you are comfortable with Mathematica or some other language.

      • Mathematica YouTube tutorial by Miller: http://www.youtube.com/watch?v=g1oj7CIqGM8

      • Mathematica  template to download: http://web.williams.edu/Mathematics/sjmiller/public_html/math/LaTexMathematica/MathematicaIntroVer6.nb (note: this was written awhile ago and some of the commands may or may not run, depending on what version you're using).

    • Homework: To receive full credit for the Project Euler portion of the course you must have 15 problems solved if  you have worked in groups of 4 or less, 19 if you have worked in groups of 5 to 8, and 27 if you have worked in groups larger than 9). You are on the honor system to have the code done, so when we meet you will be able to show it to me and discuss the ideas behind it and have it run. (If you are doing massive numbers of Putnam problems instead please let me know). Remember part of the course grade is described as a presentation; I want to meet with people at your convenience during reading period / exam period to get a good data point on what you've done and learned, and that will be an excellent opportunity. If you would rather show some of your work to the entire class, that's an option and just let me know. These problems ideally will be done and ready to show me by the start of reading period; if you need more time you need to let me know ASAP.

    • Homework: Monday's class will be split like this again.

• Week 12: Nov 17 to 21, 2014:

  • Return of Project Euler: https://projecteuler.net/

    • Make sure you have a Project Euler account.

    • Make sure you are comfortable with Mathematica or some other language.

      • Mathematica YouTube tutorial by Miller: http://www.youtube.com/watch?v=g1oj7CIqGM8

      • Mathematica  template to download: http://web.williams.edu/Mathematics/sjmiller/public_html/math/LaTexMathematica/MathematicaIntroVer6.nb (note: this was written awhile ago and some of the commands may or may not run, depending on what version you're using).

    • Homework: Solve or significantly attack 10 Project Euler problems you have not done before (unless you've already done 50 in which case your previous work has paid off!). If you work in groups, groups of up to 4 may do 10, groups of size 5 to 8 must do 14, groups of size 9 to 12 must do 18, groups of size 13 to 18 must do 22, and groups larger than 18 must do 25.

• Week 11: Nov 10 to 14, 2014:

  • Read: The 2008 Green Chicken Exam for Wednesday: Be prepared to Discuss in Class: http://web.williams.edu/Mathematics/sjmiller/public_html/greenchicken/exams/GreenChickenExamContest2008_Final.pdf

  • Remember the Green Chicken exam is Friday (Bronfman 106, give or take) around 10 or 10:30am till noonish.

• Week 10: Nov 3 to 7, 2014:

  • Read: Read the two player games handout (http://web.williams.edu/Mathematics/sjmiller/public_html/331/Math331_TwoPlayerGames.pdf), in particular be prepared to talk about 4, 5 and 6 in class on Wednesday.

  • Read: Read the textbook for the course on the Josephus problem and be prepared to talk about that in class on Wednesday.

  • Written HW: One way to prepare for math contests is to write your own contest problem! I will try to help you get practice and have your problems considered for the AMC high school or junior high school math competitions. In groups of up to N people, write up to N problems (the number of problems equals the number of people in your group!). These are to be emailed to me by 11am this Friday. They should be multiple choice (5 options), and should include a statement of the problem, the possible answers, and a solution. You can find databases here: http://www.artofproblemsolving.com/Wiki/index.php/AMC_Problems_and_Solutions

  • Project Euler: We'll pivot more to Project Euler after the Green Chicken (Saturday November 15); you might want to start looking at some of these problems again and think about coding.

• Week 9: Oct 27 to Oct 31, 2014:
  • Homework: send in some logic problems / interview problems for discussion....
  • Logic Riddles: Suggestions from class (some of these were found on the web and I've just used different phrasings).
    • Gonna make your brown eyes blue: http://mathriddles.williams.edu/?p=80
      • 1,000 mathematicians live on an island. 600 have brown eyes, and 400 have blue eyes. According to an ancient tradition which is honored by all, if a
        person ever discovers that he has blue eyes, he will promptly commit suicide the following morning. There are no mirrors on the island, and it is considered to be in very
        poor taste to discuss the subject of eye color. Of course, everyone knows what everyone elses eye color is. One day, a soothsayer passes through the island, and in the presence of the assembled inhabitants solemnly announces: At least one of you on this island has blue eyes. What happens? What information was divulged by the soothsayer that wasnt common knowledge before? Extra credit: suppose each of the blue-eyed inhabitants had resolved not to commit suicide if they ever discovered their eye color. What happens now?
    • You light up for life: http://mathriddles.williams.edu/?p=146
      • A warden meets with 23 prisoners. He tells them the following: Each prisoner will be placed into a room numbered 1-23. Each will be alone in the room, which will be soundproof, lightproof, etc. In other words, they will NOT be able to communicate with each other. They will be allowed one planning session before they are taken to their rooms. There is a special room, room 0. In this room are 2 switches, which can each be either UP or DOWN. They cannot be left in between, they are not linked in any way (so there are 4 possible states), and they are numbered 1 and 2. Their current positions are unknown. One at a time, a prisoner will be brought into room 0. The prisoner MUST change one and only one switch. The prisoner is then returned to his cell. At any time t, given some N>0, there exists a finite t_0 by which time every prisoner will have visited room 0 at least N times. (In other words, there is no fixed pattern to the order or frequency with which prisoners visit room 0, but at any given time, every
        prisoner is guaranteed to visit room 0 again. If youre still confused by this statement, ignore it, and you should be ok). At any time, any prisoner may declare that all 23 of them have been in room 0. If right, the prisoners go free. If wrong, they are all executed. What initial strategy is 100% guaranteed to let all go free? (Of all the riddles I've posted on my page, this one caused me the most difficulty.)
    • Grid games:
      • Two players are presented with an arrangement of points on a Cartesian  coordinate lattice. On each player's move, they may take away any nonempty (but not necessarily contiguous) subset of the points in a line parallel to one of the axes. Whoever cannot move loses. What can you say for an \(m \times n\) board? Go to http://www.reddit.com/r/mathriddles/comments/2freeb/a_combinatorial_game_crosspost_rmathproblempms/ for a picture depiction of a game
    • Egg drop:
      • A building has 100 floors. One of the floors is the highest floor an egg can be dropped from without breaking. If an egg is dropped from above that floor, it will break. If it is
        dropped from that floor or below, it will be completely undamaged and you can drop the egg again. Given two eggs, find the highest floor an egg can be dropped from without breaking, with as few drops as possible.
  • Homework: Due Friday, October 31: 2010 Green Chicken exam: http://web.williams.edu/Mathematics/sjmiller/public_html/greenchicken/examsolns/GreenChickenQs_2010.pdf
• Week 8: Oct 20 to Oct 24, 2014:
  • Homework: Due Monday, October 20: Skim the differentiating under the integral sign identity: http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf
  • Homework: Due Monday, October 20: I don't know why the integration handout printed poorly, but here it is: MSWord version    PDF version  Please take a look at some of the problems.
  • Homework: To be discussed in Wednesday, October 22: Do problems 1 and 3 of the Rochester 2008 Contest: http://www.math.rochester.edu/people/faculty/dangeba/2008Olympiad.pdf
    • Logic Riddles: Suggestions from class (some of these were found on the web and I've just used different phrasings).
      • Gonna make your brown eyes blue: http://mathriddles.williams.edu/?p=80
        • 1,000 mathematicians live on an island. 600 have brown eyes, and 400 have blue eyes. According to an ancient tradition which is honored by all, if a
          person ever discovers that he has blue eyes, he will promptly commit suicide the following morning. There are no mirrors on the island, and it is considered to be in very
          poor taste to discuss the subject of eye color. Of course, everyone knows what everyone elses eye color is. One day, a soothsayer passes through the island, and in the presence of the assembled inhabitants solemnly announces: At least one of you on this island has blue eyes. What happens? What information was divulged by the soothsayer that wasnt common knowledge before? Extra credit: suppose each of the blue-eyed inhabitants had resolved not to commit suicide if they ever discovered their eye color. What happens now?
      • You light up for life: http://mathriddles.williams.edu/?p=146
        • A warden meets with 23 prisoners. He tells them the following: Each prisoner will be placed into a room numbered 1-23. Each will be alone in the room, which will be soundproof, lightproof, etc. In other words, they will NOT be able to communicate with each other. They will be allowed one planning session before they are taken to their rooms. There is a special room, room 0. In this room are 2 switches, which can each be either UP or DOWN. They cannot be left in between, they are not linked in any way (so there are 4 possible states), and they are numbered 1 and 2. Their current positions are unknown. One at a time, a prisoner will be brought into room 0. The prisoner MUST change one and only one switch. The prisoner is then returned to his cell. At any time t, given some N>0, there exists a finite t_0 by which time every prisoner will have visited room 0 at least N times. (In other words, there is no fixed pattern to the order or frequency with which prisoners visit room 0, but at any given time, every
          prisoner is guaranteed to visit room 0 again. If youre still confused by this statement, ignore it, and you should be ok). At any time, any prisoner may declare that all 23 of them have been in room 0. If right, the prisoners go free. If wrong, they are all executed. What initial strategy is 100% guaranteed to let all go free? (Of all the riddles I've posted on my page, this one caused me the most difficulty.)
      • Grid games:
        • Two players are presented with an arrangement of points on a Cartesian  coordinate lattice. On each player's move, they may take away any nonempty (but not necessarily contiguous) subset of the points in a line parallel to one of the axes. Whoever cannot move loses. What can you say for an \(m \times n\) board? Go to http://www.reddit.com/r/mathriddles/comments/2freeb/a_combinatorial_game_crosspost_rmathproblempms/ for a picture depiction of a game
      • Egg drop:
        • A building has 100 floors. One of the floors is the highest floor an egg can be dropped from without breaking. If an egg is dropped from above that floor, it will break. If it is
          dropped from that floor or below, it will be completely undamaged and you can drop the egg again. Given two eggs, find the highest floor an egg can be dropped from without breaking, with as few drops as possible.
  • Homework: Due Friday, October 24: 1993 Green Chicken exam: http://web.williams.edu/Mathematics/sjmiller/public_html/greenchicken/exams/gcexam1993.pdf
• Week 7: Oct 13 to Oct 17, 2014:
  • No written HW due this week --  you should spend the time working on problems, reading the material, ....   
  • For Friday's class come prepared to talk about the following problem; have your answer for at least the first variant. First variant: You are given a number such as 100 (or, more generally, \(S\)) and you want to write it as a sum of positive integers such that the product of these integers is as large as possible; thus \(a_1 + \cdots + a_n = 100\) and we want \(a_1 a_2 \cdots a_n\) to be as large as possible (again, can do with \(S\) instead). How should we divide? Note that you are not told ahead of time what \(n\) should be. Second variant: what if instead of requiring the parts to be integers we only require them to be positive real numbers?
  • For Friday's class: we'll do more on integration. I don't know why the handout printed so poorly, but here it is: MSWord version        PDF version  Please take a look at some of the problems.
  • For Friday's class: think about which is bigger: \(e^\pi\) or \(\pi^e\). It is possible to determine which is larger without using a calculator. Can you do it?
• Week 6: Oct 6 to Oct 10, 2014:
  • Homework: Due Wednesday, Oct 8: Read / skim the following on mono-invariants / invariants:
    • Slides on talk on Invariants, Monoinvariants and Extrema: http://www.mit.edu/~pengshi/math149/talk2.pdf
    • Berkeley Math Circle Lecture Notes on Monoinvariants: http://mathcircle.berkeley.edu/BMC3/monovar.pdf
    • Problems on invariants and monoinvariants: http://math.stanford.edu/~vakil/putnam07/07putnam5.pdf
  • Homework: Due Friday, Oct 10: http://web.williams.edu/Mathematics/sjmiller/public_html/greenchicken/exams/gcexam1992.pdf (You must try all the problems on your own for at least 10 minutes before you discuss with anyone else).
• Week 5: Sept 29 to Oct 3, 2014:
  • Homework for Monday: Look at http://www.eecs.berkeley.edu/~oholtz/H90/hmw3.pdf and try the problems we haven't done in class and be prepared to discuss on Monday.
  • Homework for Wednesday: Try the 2012 Green Chicken problems (http://web.williams.edu/Mathematics/sjmiller/public_html/greenchicken/exams/GreenChickenQs_2012ExamSolns.pdf) and come to class with questions or comments about the problems you find interesting. Note the Green Chicken is famous: http://math.williams.edu/boston-com-pictures-miller-with-chicken/ (story from Forbes Magazine)
  • Homework for Wednesday: Read about countable and uncountable:
    • http://www.math.psu.edu/wysocki/M403/Notes403_3.pdf
    • http://www.math.uiuc.edu/~hildebr/347.summer14/cardinality.pdf
  • Homework: Due Friday, Oct 3: #1: Find the final digit (i.e., the ones digit) of 2^(3^(4^5)). #2: Prove 2^(n-1) divides n! if and only if n is a power of 2. #3: How many primes are there such that, if the prime is written in base 10, its digits are an alternating string of 0s and 1s with first digit and last digit 1? #4: Show that if n divides a Fibonacci number that it divides infinitely many Fibonacci numbers. #5: For all positive real numbers a, b, c show that \(a^a b^b c^c \ge a^b b^c c^a\) . #6: Show that if a, b and c are positive numbers summing to 1 that \((a + 1/a)^2 + (b + 1/b)^2 + (c + 1/c)^2\) is at least 100/3. HW  is still due on Friday even if Mountain Day.
  • Homework: Keep looking at problems in the book, online, ... and find topics you would like to see.
• Week 4: Sept 22 to 26, 2014:

  • Read: From the textbook (Famous Puzzles): Pages 11-13, 201-202 (for the brave: derive (7.10)), 296-297.

  • Read: Generating Functions Handout: This is from a book I'm writing on probability. The first section is motivation, feel free to skim. Section 19.2 is the most important. Section 19.3  is more technical and included for completeness; we won't cover. Section 19.4 talks about convolutions -- we'll need the very beginning to analyze the Catalan numbers.

  • Read: Recurrence Relations Handout: Goes through the analysis of the double plus one strategy from roulette (this was the video I made with students from OIT).

  • Read: Recurrence Relations Handout Part 2: Goes through the algebra to solve recurrences.

  • Read some probability problems and some number theory problems; try a few and let me know which you want to discuss.

  • Homework: Due Friday, Sept 26: #1: How many ways are there to cover a \(3 \times n\) board using just \(1 \times 2\) tiles? #2: What if  now we have a \(2 \times 2 \times n\) box and just \(1 \times 1 \times 2\) tiles? #3: Solve the double recurrence \(f_n = f_{n-1} + 3 g_{n-1}, g_n = -3 f_{n-1} + 9 g_{n-1}\). \#4: Define a set to be selfish if it contains its cardinality (i.e., its number of elements) as an element; thus {1,3,5} is selfish, while {1,2,3,5} is not. Find, with proof, the number of subsets of {1,2,...,n} that are minimal selfish sets (that is, selfish sets none of whose subsets are selfish; thus {1,3,5} is not minimal selfish as {1} is a subset). This is a Putnam problem.....

  • Homework: Solve four project Euler problems (if you've already solved many you're fine!). You may do these in groups, you do not need to hand them in, as always feel free to talk to me. I want you to build up your programming skills if they're not well-developed, and see how powerful a tool it can be.

  • Homework: Tell me what topics you would like to see. Look through our book for things that you find fun. One option is to run Friday's class the way I do math puzzle night; I'll bring in a random math competition from somewhere in the world and we'll analyze and attack the problems together.

• Week 3: Sept 15 to 19, 2014:

  • Reading for Friday: read about Recurrence relations.

    • Baby bear introduction to solving recurrences: https://www.cs.duke.edu/~reif/courses/alglectures/skiena.lectures/lecture3.pdf

    • Mama bear: http://www.webpages.uidaho.edu/~markn/395/pdf/rec-eq.pdf

    • Papa bear: http://www.cs.columbia.edu/~cs4205/files/CM2.pdf

    • Video I made with OIT about applications of recurrence relations to gambling in Vegas: https://www.youtube.com/watch?v=Esa2TYwDmwA

    • Think about how the n blue and n red dots on a circle problem, and about how many ways we can have multiple ways to walk.

  • Homework: Due Friday, September 19 (note additional problems may be added): #0: Go to Project Euler (https://projecteuler.net/) and create an account for yourself, and solve the first problem. #1 to #4: Look at the problems above, and choose four that you find interesting and solve. It is important to learn how to generalize a problem; you want to get into this habit. You can choose these four problems from the links or from Project Euler, and do not need to hand anything in. First problem to be submitted for grading is #5. Looking at the problems on covering a \(2 \times n\) board with \(1 \times x\) dominoes, there are a lot of possibilities: #5: How many ways are there to cover a \(3 \times n\) board using just \(1 \times 2\) tiles? #6: What if  now we have a \(2 \times 2 \times n\) box and just \(1 \times 1 \times 2\) tiles? #7: Choose at least one induction problem and at least one AM-GM problem to think about; you do not need to write about it. Email me if you'd like me to do it in class.

  • You should know LaTeX and Mathematica (or another programming language): tutorials are here: http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm

  • Wednesday's class: http://youtu.be/nKatBilIhmw

• Week 2: Sept 8 to 12, 2014:

  • Read up on induction. There are lots of good webpages with induction problems:

    • http://www.math.ucla.edu/~radko/circles/lib/data/Handout-142-159.pdf

    • http://math.stackexchange.com/questions/294677/looking-for-induction-problems-that-are-not-formula-based

    • http://www.math.cornell.edu/~mec/2008-2009/ABjorndahl/ppmi.pdf

    • http://staffhome.ecm.uwa.edu.au/~00021149/Academy/1995/inductionprobs.pdf

  • Homework: Due Friday, September 19 (note additional problems may be added): #0: Go to Project Euler (https://projecteuler.net/) and create an account for yourself, and solve the first problem. #1 to #4: Look at the problems above, and choose four that you find interesting and solve. It is important to learn how to generalize a problem; you want to get into this habit. You can choose these four problems from the links or from Project Euler, and do not need to hand anything in. First problem to be submitted for grading is #5. Looking at the problems oncovering a \(2 \times n\) board with \(1 \times x\) dominoes, there are a lot of possibilities: #5: How many ways are there to cover a \(3 \times n\) board using just \(1 \times 2\) tiles? #6: What if  now we have a \(2 \times 2 \times n\) box and just \(1 \times 2\) tiles? #7: Choose at least one induction problem and at least one AM-GM problem to think about; you do not need to write about it. Email me if you'd like me to do it in class.

  • Second class: Induction: http://youtu.be/ow8nGYDYbzE    Third class: AM-GM:  http://youtu.be/QgxP7l3BRuY    Fourth Class (via YouTube): https://www.youtube.com/watch?v=XwnzWOc3_-0&feature=youtu.be

• Week 1: Sept 1 to 5, 2014: First class: Mechanics: http://youtu.be/tVLPZgRt9ug (plus a bit on Pascal's triangle)
  • Welcome slides for first class.
  • Handout for first class.
  • Homework: To be emailed to me by 8pm on Sunday, September 7: Email me a short note (a paragraph suffices) on what you want to get out of this course, and what lesson  you learned from the graduation speech by Uslan (http://www.graduationwisdom.com/speeches/0018-uslan.htm). Full credit, 20/20, so long as you answer both questions on time.
  • Homework: To be done by start of class on Monday, September 8: Read this short note on proof techniques: http://web.williams.edu/Mathematics/sjmiller/public_html/331/handouts/proofs.html
  • Homework: To be done by start of class on Monday, September 8: Read the pages here on induction: http://web.williams.edu/Mathematics/sjmiller/public_html/331/handouts/AppendixI.pdf
  • Homework: To be done by start of class on Wednesday, September 10: Read this short note on the AM-GM: http://web.williams.edu/Mathematics/sjmiller/public_html/331/handouts/ArithMeanGeoMean.pdf
  • Homework: To be done by start of class on Monday, September 8: Try a few problems from here: http://math.stanford.edu/~vakil/putnam07/07putnam1.pdf