Fall 2011 quarter  9/25/2011  We will get to know each other and solve a variety of problems. Handouts: Welcome Handout Far away, in a distant galaxy, there is hotel called Hotel Infinity. True to its name, the hotel has infinitely many rooms. Currently, all the rooms are taken. Then, a guest arrives. Can you find a space for the new guest? What if 10 guests arrive? Can you combine two such hotels into one?
In this first meeting related to our studies of infinity, we will solve a series of problems related to Hotel Infinity. Handouts: Hotel Infinity We will investigate the continued fraction expansions, an alternative format to decimal notation for representing a number which reveals different types of information about that number's properties. Handouts: Continued Fractions We will look at representations of numbers as fractions and decimals. Is every fraction a decimal? Is every decimal a fraction? How can you tell? How can you convert between the two? What are the properties of each type of representation, and what relationships exist between equivalent representations of the same number? Handouts: FractionsAndDecimals1  FractionsAndDecimals1Answers In this 6hourslong (3 meetings) minicourse, we shall compare a few 2dimensional (2D) surfaces, the Euclidean plane, a cylinder, a Moebius plane (a.k.a. an unbounded Moebius strip), a sphere, and, if time permits, the Lobachevsky plane.
Handouts: Geometry I We will solve a variety of problems that use the concept of parity in sometimes unexpected ways. Handouts: Parity  I  10/2/2011  We will be doing some or all of the following:
1. Picture Coding
2. A riddle word problem
3. Roman Numerals exercise
4. "How much does it weigh?" worksheet involving a balance
(A worksheet will be posted later in the week) Handouts: Balance scale handout We will continue the study of infinity and working with problems related to the special Hotel Infinity. We will be expanding this idea of infinity to the Infinity Rockets, ultimately leading up to finding an algorithm of how to fit infinitely many people into a rocket with infinitely many seats. Handouts: Hotel Infinity II This week we will try to identify and explain patterns in the continued fraction expansions of numbers. Handouts: Continued Fractions, Ctd. We will continue what we began last week, looking at representations of numbers as fractions and decimals. Is every fraction a decimal? Is every decimal a fraction? How can you tell? How can you convert between the two? What are the properties of each type of representation, and what relationships exist between equivalent representations of the same number? Handouts: FractionsAndDecimals2  FractionsAndDecimals2Answers In this 6hourslong (3 meetings) minicourse, we shall compare a few 2dimensional (2D) surfaces, the Euclidean plane, a cylinder, a Moebius plane (a.k.a. an unbounded Moebius strip), a sphere, and, if time permits, the Lobachevsky plane.
Handouts: Geometry II We will continue our study of parity started last week. Handouts: Parity  II  10/9/2011  This week we will be working again with balances but children will need to use reasoning to find a fake coin among a group of real coins with the help of the balance. They need to be able to understand the difference between worst/best case scenario and how the fake coin will not always be in the same place. Handouts: Find the Fake Coins We will transition from the idea of infinity into sets. More specifically, ideas including maps between sets, onetoone, onto and equivalence of sets. Handouts: Sets and Functions This week we will finish our discussion of continued fractions. We will continue working on the handout from last week, along with a few more problems. Basic calculators are again encouraged for this week's session. If you have time during the week, get in a little practice performing the computations we've discussed so far! We will continue our series on the relationship between fractions and decimals, focusing on how the period of a repeating decimal works. UPDATE: The homework for students is to complete problems 17 of the attached handout before next week, if they have not already done so. Handouts: FractionsAndDecimals3  FractionsAndDecimals3Answers In this 6hourslong (3 meetings) minicourse, we shall compare a few 2dimensional (2D) surfaces, the Euclidean plane, a cylinder, a Moebius plane (a.k.a. an unbounded Moebius strip), a sphere, and, if time permits, the Lobachevsky plane.
Handouts: Geometry III This week we will start a study of combinatorics, the art of counting arrangements or combinations of objects. Handouts: CombinI  10/16/2011  We will learn another way to write numbers and will solve many problems using Roman numerals Handouts: Roman Numerals  Fix addition problems We will be continuing with sets and functions. Handouts: Sets and Functions II This week Mike's group will begin a new topic: Classical compass and straightedge constructions. Handouts: Construction problems We continue our series on fractions and decimals. (Continuing with previous session's handout [see 10/09 meeting description].) A regular ngon is a polygon with n sides of equal length and all angles between
adjacent sides equal. An alternative description is that all corners of a regular ngon lie
on a fixed circle, and the angle between two adjacent corners, as seen from the center
of the circle, is pi/n .
The theme of this session is to construct different regular ngons. Handouts: Ruler and Compass Construction We will continue our discussion of combinatorics started last week. Handouts: CombinII  Combin ChallengeII  10/23/2011  We will now use the idea of a 11, onto function to analyze infinite sets. Handouts: Infinite Sets This week Mike's group will look at some more of the constructions from last week, as well as connections with arithmetic. Having thoroughly picked apart the relationship between fractions and decimals, we'll turn our attention to a generalization of repeating decimalsthe fabled geometric series. Handouts: GeometricSeries In this talk, we will continue with constructing regular ngon using straight edge and compass. Then we will explore the possibility of construction using only compass. Handouts: Mohr–Mascheroni Theorem We will continue our discussion of combinatorics started two weeks ago. Handouts: Combin  III  10/24/2011  Handouts: Meeting 5  10/30/2011  Handouts: Young Diagrams We will be learning different ways to multiply numbers and comparing these methods to regular long multiplication. Handouts: Egyptian Multiplication Following the high school group, this week Mike's group will investigate the construction with compass and straightedge of regular polygons in the plane, and connections with algebra.
For more information on constructing regular polygons, see:
http://en.wikipedia.org/wiki/Constructible_polygon Handouts: Regular Polygons We'll wrap up geometric series and start a brand new topic: Parity!
Update 10/31: Try to complete problems 1 and 2 of the Parity1 handout before next week. (It's not necessary to complete the others, as there will be time to work on them during the next session.) Handouts: Parity1 In this talk, we will study various uses of the pigeonhole principle. Handouts: Pigeonhole principle This week we start learning about some basic number theory. In particular, we talk about prime numbers, prime factorization, and divisibility.
Handouts: Divisibility  I  11/6/2011  The Tower of Hanoi or Towers of Hanoi , also called the Tower of Brahma or Towers of Brahma, is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.
The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:
* Only one disk may be moved at a time.
* Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
* No disk may be placed on top of a smaller disk.
Handouts: Hanoi Tower Part 1  We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2. Handouts: Russian Peasant Multiplication Mike's group continue with last week's handout on constructing regular polygons with compass and straightedge.
For more information on constructing regular polygons, see:
http://en.wikipedia.org/wiki/Constructible_polygon Our look at problems involving parity (evenness/oddness) continues. Handouts: Parity2 Many amazing constructions can be accomplished using straightedge and compass. However, one cannot trisect arbitrary angle or construct cube root 2. In this talk and the next one, we will look at simple ways to execute the above constructions by folding papers, and consider the possibilities when different folding moves are allow. Handouts: Paper_folding_construction_1 This week, we continue our discussion from last week. Handouts: Divisibility  II  11/13/2011  The Hanoi Tower is a type of algorithm, which is a method used to give a finite list of detailed instructions. Last week, we allowed the children to play with the puzzles but this week we will teach them how to correctly record their steps and movements that way the puzzle can be solved correctly and efficiently. Handouts: Hanoi Tower Part 2 We will use both Egyptian and Russian Peasant Multiplication to transition into binary notation. This includes writing numbers in binary, adding them and multiplying them. Handouts: Binary Notation This week Mike's group will investigate the effects of a different notion of distance in the plane on familiar geometrical objects and facts. Handouts: Taxicab Geometry We will wrap up our sessions on parity.
(Note: The two parity handouts are posted in the previous two session descriptions.) We will continue from last week by constructing cube root of 2 and explore the connection between folding papers and parabolas. Handouts: Paper_folding_construction_2 In this session we introduce pigeon hole principle, which is an important problem solving technique. Handouts: Pigeon hole I  11/20/2011  This week we will be doing a worksheet called cutting logs, which will involve the kids learning how to make appropriate patterns when given an input (number of cuts) and determining the output (number of pieces);this can also be reversed. They will then be asked to put the pattern in an equation form. For example, if I have x cuts, how many pieces, y, will i have? How can I always know the answer without actually have to draw all the cuts? Handouts: Cutting Logs We will be continuing with Binary Notaion for the week. Please make sure your child brings the worksheet back to Math Circle for this week. This week Mike's group will investigate taxicab geometry further, continuing last week's discussions of the analogues of conic sections, and comparing taxicab geometry to the axioms of Euclidean geometry. Handouts: Axioms of Geometry We will work together on some problems about working together. Handouts: WorkingTogether  WorkingTogetherSolutions During the past two meetings, we used paper foldings to trisect angles and solve cubic equations in general. This week, we will change our perspective (literally) and look at parabolas in the projective plane. In particular, we will develop enough tools to find the common tangent of two parabolas in homogeneous coordinate, the coordinate system of the projective plane. Handouts: Projective Geometry This week, we continue our discussion from last week. Handouts: Pigeon hole II  11/27/2011  In observance of Thansgiving, there will be no Math Cirlce this week. Handouts: Extra (1)  Extra (2)  Extra (3)  12/4/2011  We will be doing some fun problem solving questions, with some review questions thrown in of course to end the quarter. Please join us for a fun team problem solving competition. There will be prizes for the top teams. This is our last meeting of the fall quarter. For the last meeting of the year, we continue the tradition of math relays, in which students form teams and race to solve the most problems.  Winter 2012 quarter  1/15/2012  This week, we will be working with triangles and will be going over a couple of review problems from last quarter. Handouts: WarmUp  Handout 1 Hidden in the jungle near Hanoi, the capital city of Vietnam, there exists a Buddhist monastery where monks keep constantly moving golden disks from one diamond rod to another. There are 64 disks, all of different sizes, and three rods. Only one disk can be moved at a time and no larger disk can be placed on the top of a smaller one. Originally, all the disks were on one rod, say, the left one. At the end, they all must be moved to the right rod. When all the disks are moved, the world will come to an end. (No worries here, it will take the monks a few hundred billion years to complete the task.)
In this session, we shall first play with the puzzle and try to figure out the fastest way to solve it. Next, we shall study some auxiliary material needed to better understand the puzzle. This includes placevalue numeral systems, like the decimal system we use for counting, the binary system that formed the bedrock of Egyptian multiplication we have learned in the Fall quarter, and pizza slicing as a way to think about fractions.
Mike's group will begin the new year by looking at knots! In the first session of the new year, we'll start exploring the mathematics of making (and breaking!) secret codes.
UPDATE 01/15/2011: For help with solving simple substitution ciphers like the last one on this handoutfor example, a list of letter frequencies in English text (and more)see the Frequency Analysis entry on Wikipedia: http://en.wikipedia.org/wiki/Frequency_analysis. Handouts: CodesAndCiphers1 We will discuss modular arithmetic, the Chinese Remainder Theorem, Euler's Theorem, Wilson's Theorem, and then work through a variety of olympiadstyle number theory problems. Handouts: Number_theory_handout_I This week we are introducing the notion of graphs and learn about some of their properties.
Handouts: Graph I  1/22/2012  Handouts: Meeting 2 Handout We shall find an optimal algorithm solving the Hanoi tower puzzle for any number of disks. Handouts: Class handout. This week Mike's group will explore more examples of knots, and begin to discuss the mathematical theory of knots. We will continue our study of cryptology with a look at the Vigenere cipher, once believed by many to be unbreakable.
NOTE: For homework, complete sections 14. On Section 5, begin by numbering the cipher (the first one, which begins "KVX DOGRUXI OM R PHRHKVBMRZ ...") in 123123123 fashion, and since there is a single letter R, guess either R=a or R=i for the ciphertext letters under number 1. Use your guess to decode the first three lines, but only the letters under number 1. We'll work on it more next week. Handouts: CodesAndCiphersFull We will discuss modular arithmetic, the Chinese Remainder Theorem, Euler's Theorem, Wilson's Theorem, and then work through a variety of olympiadstyle number theory problems. Handouts: number_theory_handout_II This week, we will continue our discussions on graphs from last week. Handouts: Graph II  1/29/2012  Handouts: Meeting 3 Handout We shall have a Hanoi Tower contest at the beginning of the class. The first student to assemble the puzzle with six disks will be pronounced the Great Master of Disks and will get a $1 reward. We shall proceed to solve the problems form the second handout we haven't solved yet. We shall next study the decimal placevalue numeral system currently in use by humanity. Handouts: This week Mike's group will discuss geometric and other kinds of objects in dimensions greater than three. Having looked at some of the classical ciphers, we'll shift attention to the foundation of modern publickey cryptography (and a really cool area of math in its own right), modular arithmetic.
UPDATE: Homework is to completely decrypt one or both of the Vigenere ciphers in the Crypto packet (see previous week for handout). Also if you were not present in class, read the Modular Arithmetic packet up to Problem 1, and try to solve Problem 1. (We will continue work on the rest of it next meeting.) Handouts: ModularArithmeticI This week we will talk about triangle inequality, which is an important tool in solving many geometric problems. Handouts: Triangle Inequality I  2/5/2012  Handouts: Meeting 4 Handout To make sure that no student is left behind, we shall solve a few more Hanoi Tower puzzle problems. Once finished, we shall begin our study of placevalue numerals. Handouts: Digits and numbers. This week Mike's group will continue its study of higher dimensional spaces, and look at some counterintuitive aspects of spheres and cubes in high dimensions. We'll continue our exploration of modular arithmetic. The talk will illustrate two themes. The first is that topology, a branch of mathematics that was developed during the 20th century, can be viewed in part as a natural continuation of mathematicians' efforts, from the 17th century onward, to clarify the foundations of calculus. The second is that, in contrast to sciences in which a new theory supplants all previous ones, in mathematics new developments generally arise as extensions of previous discoveries. Specifically, we will trace a portion of the history of calculus and topology to see how it served as background to research conducted late in the 20th century. Handouts: Calculus, topology and fixed point This week, we continue our discussion from last week on triangle inequality and its usage in solving problems Handouts: Triangle Inequality II  2/12/2012  Handouts: Meeting 5 We shall continue working through the handout for the previous class. This week Mike's group will discuss Eves's theorem, and its application to studying pictures in perspective. We'll talk about when you can (and can't) divide in modular arithmetic, and how to do it when you can. Handouts: ModularArithmeticII We will review the problems and their solutions from the 2012 AMC 10/12A. This week, we talk about the notion of winning strategy and play some mathematical games! Handouts: Game Theory I  2/19/2012  Cryptarithmetic, also know as cryptarithm, alphametics, or word addition, is a math game of figuring out unknown numbers represented by words. Different letters correspond to different digits. Same letters correspond to same digits. The first digit of a number cannot be zero. Deciphering cryptarithms is a great way to further familiarize ourselves with our numeral system (decimal placevalue). Handouts: Cryptarithms. This week Mike's group will continue with last week's discussion of looking at geometric objects in perspective. Handouts: Perspective We'll take a fond look at one of the world's oldest algorithms, the Euclidean Algorithm, and explore its relationship to modular arithmetic.
UPDATE: Homework is to complete the Euclidean Algorithm handout. Handouts: EuclideanAlgorithm We will go over several problems from AMC10/12 competitions with beautiful, cool solutions. Handouts: Cool AMC Problems This week we will start learning about the powerful tool that is mathematical induction. Handouts: Induction I  2/26/2012  In this class, we shall study geometric series. If time permits, we shall study the Sierpinski carpet and employ a geometric sequence to figure out its area. Mike's group will apply our explorations of projective goemetry to investigating algebraic curves in the plane. Handouts: Homogeneous Coordinates We'll discuss the Extended Euclidean Algorithm and how it relates to inverses in modular arithmetic. Handouts: EuclideanAlgorithm2 Today we continue our discussion of induction. Handouts: Induction II  3/3/2012  Mike's group will learn to compute the intersection number of two curves which meet at the origin in the plane. Handouts: Intersection Number Exercises  3/4/2012  In this class, we will study some remarkable features of the Sierpinski triangle. At the end of the class, we will take a look at other fractals, geometric objects of fractional dimensions (hence the name), more from the point of view of art rather than science. We'll continue our investigation of number theory topicsthis timem, more on modular inverses and Euler's Phi function. Handouts: EuclideanAlgorithmAndEulerPhi Groups are important algebraic structures which are used in all areas of mathematics, as well as physics and chemistry. The goal will be to define and give some examples of these objects. The main example will be the braid groups, which is a mathematical formulation of what happens when we braid our hair. Handouts: Braids and Groups Today we finish our discussion of induction with some word problems involving induction. Handouts: Induction III  3/11/2012  Mike's group will investigate Euclid's algorithm for polynomials, reviewing the standard Euclid algorithm as well as long division for polynomials. We will also continue our investigation of intersection numbers. Handouts: Euclid's Algorithm, Polynomial Long Division In preparation for RSA, we do a little more digging on Euler's phi function and raising numbers to powers modulo another number. Handouts: EulerModular A rational triangle is a right triangle whose three sides are all rational numbers. A rational number is called a congruent number if it is the area of some rational triangle. It is an ancient problem to decide if an arbitrary integer is a congruent number. Surprisingly, this innocentlooking problem is still open today. In this talk, we will go through the history of congruent numbers and look at its connections to important objects in modern number theory. Handouts: congruent number handout  congruent number handout answer This week we will start discussing propositional logic. Handouts: Prop. Logic I  3/18/2012  Today will be a review of all that we have covered this quarter. We will be keeping the exams to review the responses of each child to help better understand what to do next quarter, but I encourage you go over the questions with you children! Handouts: Meeting 10 In this class, we will train for the upcoming Math Kangaroo competition by solving some harder problems from the previous years' events. In this class, we will see how states of the Hanoi Tower puzzle can be represented by graphs very similar to the ones formed by the vertices and sides of the Sierpinski triangle approximations. See the handout for more. Handouts: class handout Returning one more time to cryptography, we put all the number theory to use and take a look at the publickey RSA cryptosystem. Handouts: RSAHandout Today we continue our discussion of propositional logic started last week. Handouts: Prop. Logic II  Winter 2012 quarter  3/25/2012   4/1/2012   Spring 2012 quarter  4/8/2012  To start the quarter off, we will start work with binary notation. We will introduce this concept first by going back to the previous lessons of balancing of scales. Handouts: Handout 1  Solutions 1 In this class, we will learn how geometry has begun, proceed to Euclid's "Elements" and discuss the notions of an axiom, a point, line, and straight line. At the end, we will do some Greekstyle geometry using a rope to perform multiplication and even raising a number to a power! Handouts: handout Mike's group will begin the Spring Quarter with a review of basic modular arithmetic, as well as a discussion of modular arithmetic for polynomials. Handouts: Some Proofs in Modular Arithmetic  Modular Arithmetic for Polynomials  Practice Computing Residues The distance between two points is the length of the shortest line connecting them. Normally, such a path is a straight line. But in a city consisting of a square grid of streets shortest paths between two points are no longer straight lines (as every cab driver knows). We will explore the geometry of this unusual distance and play several related games.
Handouts: TaxicabGeometry We are all familiar with the absolute value function. It measures the distance of a number from 0. We will examine the essential properties of this function, define other absolute values on the rational numbers, and classify all such absolute values. This leads to the idea of completions and the padic numbers which we discuss briefly. Finally we introduce Hensel's Lemma for solving polynomial equations over the padics and mention the localglobal philosophy. Handouts: Absolute Values Handout Today we will solidify our understanding our material covered earlier this year in preparation for the rest of the quarter. Handouts: Review  4/15/2012  This week we will be continuing to work with binary notation. We will show how there are similarities and differences between binary notation and decimal notation place values. We will also start to work with adding and subtracting binary numbers. Handouts: Handout 2 In this class, we will learn various ways to construct a triangle with a compass and ruler. Handouts: handout Mike's group will discuss bounded and unbounded sets, and the notion of a least upper bound. Handouts:  Boundedness The concept of proof lies at the heart of mathematics, yet school mathematics usually spends very little time discussing them. What is a proof? Why do we need them? How do you write them, or verify someone else's? This week begins a series which will examine these issues. Handouts: Proofs1 The "Occupy Wall Street" movement was concerned with the inequality of income in the U. S. Are the rich in the U. S. really getting richer and the poor getting poorer? How does the U. S. compare with other countries in this regard? The Gini Index is an important statistical tool for answering such questions. You will use geometry and some calculus to study its mathematical properties. In particular, we will investigate some mathematical models of income distribution and the corresponding Gini indices. This week we will begin studying modular arithmetic. Handouts: Mod. Arith. I  4/22/2012  This lesson we continue our lesson on Binary Numbers. This time we play a game of "What's my number?" with the binary tree and the card trick game. Handouts: Handout 3 In this class, we will first finish covering the parts of Handouts 1 and 2 we haven't covered yet. If time permits, we will do some more compass and ruler constructions. Handouts: handout Mike's group will investigate bounds on sets of numbers, and several important inequalities. Handouts: Inequalities We will introduce the concept of syllogism and try to reason our way through some tricky logical puzzles. Handouts: Proofs2 In this session we will explore different ways of sampling from populations. How is it done poorly, and how can we do it well? We'll talk about recent polls that give us some ideas for predicting the upcoming presidential election.
Additionally, we'll talk about sampling distributions  a sampling distribution tells us what to expect for our sample under certain constraints. Information from the sampling distribution allows us to test a research claim or to come up with specific methodology. We will use examples to illustrate the ideas.
Handouts: Samples & Sampling Distribution This week we will finish up the worksheet from last week.  4/28/2012  This week Mike's group will investigate more inequalities and boundedness of sets.  4/29/2012  Handouts: Handout 4 We will continue to learn using compass and ruler for constructing angles and triangles. We will look at some very useful tools in the proofmaker's toolkit: proof by contrapositive, and proof by contradiction.
UPDATE 04/29: Homework for next week is to complete pages 14 of the handout (Proofs3), if not done already. Handouts: Proofs3 Starting with Bayes' rule, we'll figure out the probability of winning the 1960's game show Let's Make a Deal. We'll extend the ideas to some examples where Bayesian thinking helps avoid common misunderstandings.
Finally, we will apply Bayesian ideas to statistical inference. Using prior knowledge, we will assess different estimators with an application to baseball batting averages.
Handouts: Bayes' Rule This week we will use our knowledge of modular arithmetic to study Diophantine equations. Handouts: Mod. Arith. II  5/6/2012  Handouts: Handout 5 We will take a break from our geometry minicourse, do some wizardry studies, save a prince from an evil king and help a prisoner choose between a cell with gold and a cell with a hungry tiger. Don't forget to bring your magic wands! Handouts: handout This week Mike's group will be discussing abstract notions of ordering, in relation to our earlier discussion of least upper bounds. We will finish up some topics from last week, and then examine the power and pitfall of proving things without using words. Handouts: Proofs4 At the beginning of Guy Ritchie's cult hit, Lock, Stock, and Two Smoking Barrels, the
main character, Eddy, gets trapped in a (rigged) poker game set up by the villain, "Hatchet"
Harry: while all the players are dealt one card face up, and two face down, Eddy's "down
cards" sit directly above a hidden camera.
After a few rounds of play, Eddy is eventually caught in a hand against Harry. He holds
10 of diamond face up, and 6 of spade 6 of heart face down,
while Harry's hand shows
9 of club ? ?
The ve hundred thousand dollar question (as it happened) Eddy faced was this:
What are the odds that Eddy's hand is the better one?
Come to the math circle this Sunday to nd out... and to learn how to attack similar
gamblingtype problems using probability.
Handouts: Talk abstract This week we will study some applications of binomial coefficients in combinatorics. Handouts: Combinations I  5/13/2012  Introducing Mayan Notation! Handouts: Handout 6 In this class, we will learn to measure angles with a protractor. We will also learn a few facts about angles and prove the first theorem of the course, the one claiming that angles of any triangle in the Euclidean plane add up to a straight angle. Handouts: handout This week Mike's group will look at some more examples of partial orderings, and their uses in solving problems in combinatorics and graph theory. We'll wrap up our series on proofs and proving with some further reflections on logic. Suppose that there are two hotels for numbers. The first hotel has rooms labeled 1 through n and each room can take at most one guest. If m numbers get rooms at the hotel and fill the hotel to capacity, then m must equal n. This seems quite natural, but how do we prove it? The second hotel is very strange. Last night, all the natural numbers were guests at the hotel and the hotel was filled to capacity. Tonight, all the fractions have decided to visit the hotel but the natural numbers won't leave. The hotel manager was in a panic because the hotel appeared to be overbooked. Fortunately, the hotel manager's sister is a mathematician and she told her brother not to panic. After rearranging the guests, she was able to fit in not only all the natural numbers, but all the fractions as well. How did she do it and what does this tell us about the infinite? This week, we continue our discussion from last week on combinations. Handouts: Combination II  5/20/2012  Handouts: Handout 7 Last Sunday, we have worked out the proofs of Propositions 1 and 2 with the first class (3:45  4:45) and the proof of Proposition 1 with the second class (5:00  6:00). This time, we will finish studying the proofs in the last handout. If time permits, we will begin a new topic, Clock Arithmetic. This week Mike's group will attempt to explain an interesting fact discussed last week, and other combinatorial results of a similar nature. We'll review problems from last week's individual problem solving, and then begin investigating some of the mathematics of motion and everyday physics.
UPDATE: We will continue the handout next week. For those who weren't here or didn't do so already, try to complete problems 110 (everything up to the statement of the Law of Lever). Handouts: BalancingAndFallingI Enumerative combinatorics is a field of mathematics that deals with counting the number of objects satisfying some combinatorial description. We will learn to count the number of a number of seemingly unrelated objects such as monotonic lattice paths and partitions of integers, and explore their relationships with each other. If time permits, we will also make some necklacesor rather, count the number of different ways to do so. Handouts: Enumerative Combinatorics This week students work on a set of problems individually, to test their level of understanding the material taught so far.  5/27/2012  Handouts: Handout 8 In this class, we will learn that time, like money, is a manmade concept, that our clocks and watches do not show time, but rather model the Earth's rotation around its axis, and that time travel is quite possible if you live next to a pole. Handouts: handout This week Mike's group will explore some examples of arithmetic functions of number theoretic significance. We continue our investigation of mechanical topics, looking more at levers and how the concept of center of mass can help us solve surprising problems. (Update: We won't be discussing pulleys this time after all.) Handouts: BalancingAndFalling2  BalancingAndFalling2Solutions We will first describe sentential logic, a language used to formalize
certain kinds of assertions. Then, we will state and prove the
compactness theorem of sentential logic, a result of fundamental
importance that in some sense allows us to understand infinite
collections of assertions by their finite subcollections. This will be
used to prove a corollary about graph colorings. This week we will continue our discussion combinations from two weeks ago. Handouts: Combination III  6/3/2012  Handouts: Handout 9 In this class, we will study the mod n arithmetic, that of a circle divided into n equal parts. For n = 12, 24, and 60, this is nothing else but the arithmetic of the face of a clock. Handouts: handout Mike's group will continue last week's discussion of several arithmetic functions with significance to number theory. Special guest instructor Prateek Puri will take us on a tour of some combinatorial problems. Just when you thought counting was as easy as 1,2,3... Question: What comes next in the following sequence: 13, 1113, 3113, 132113, 1113122113, ??
Answer:"RADON".
We will explore the reasonings behind this answer. This week we will review some of the materials from last quarter about graphs, and learn about some new properties of graphs. Handouts: Graph III  6/10/2012  Handouts: Handout 10 The students will be given some problems to solve on the material covered in the Spring quarter. The papers will be checked on the fly. The highestscoring students will receive citations and awards. Please bring a compass, ruler, and sharp pencils to the class. Protractors are not needed this time.
The test and the answer key are now posted here. Please see the attached files. Handouts: test  answer key Join us for a team problem solving competition. Join us for the traditional endofyear Math Relay competition! Next week we will have the relays!! 
