Breakout Catalog Last updated 6/15/14

Week 1, Morning

Mathematica I, Prof C (Silva Chang) Mathematica is a popular computer software program used by mathematicians and scientists. We will learn how to use Mathematica to perform basic mathematical operations, solve equations, create lists, and generate 2D graphics. No computer programming experience is necessary. (At the end of camp, each MathPath participant will receive a free copy of Mathematica.) 1 star Cryptology, Dr C (David Clark) Cryptology is the art of creating – and breaking! – ciphers which take information and make it secret. We'll begin by looking at basic codes, and using modular arithmetic to explain how certain ciphers can be described mathematically. We'll also examine the Vigenere Cipher, which was described as "unbreakable" just 100 years ago, but is nearly as easy for mathematicians to break as a simple "Cryptoquip" in the newspaper. By the end of the week we'll cover modern cryptosystems, including methods used by websites like Facebook and Amazon to protect passwords and credit card numbers. Knowledge of modular arithmetic, for instance from Number Theory I or the equivalent, will be used throughout the week. Additional number theory will be introduced as needed, so prior knowledge of additional number theory is helpful but not necessary. 2 stars Introduction to KenKen , Mr L (Al Lippert) Ken means clever or wisdom in Japanese. So KenKen, a puzzle invented by Tetsuya Miyamoto in 2004, means wisdom squared. As in Sudoku, the goal of each puzzle is to fill a grid with digits –– 1 through 4 for a 4!4 grid, 1 through 5 for a 5!5, etc. –– so that no digit appears more than once in any row or any column. Grids range in size from 3!3 to 9!9. Additionally, KenKen grids are divided into heavily outlined groups of cells –– often called “cages” –– and the numbers in the cells of each cage must produce a certain “target” number when combined using a specified mathematical operation. Numbers CAN be repeated in a cage. Basically, KenKen puzzles involve simple arithmetic, but solving them requires a combination of number theory, math concepts such as parity, multiple factorizations and partitions of numbers. A special technique using fault lines will be taught. Deductive reasoning, extensive use of logic AND the ability to persevere in the face of frustration are the only prerequisites for this course. Guessing will not be tolerated. 2 stars Induction, Mr M (Steve Maurer) If you line up infinitely many dominoes on their ends, with each one close enough to the previous one, and knock over the first, then all infinitely many fall down. This is the essence of mathematical induction, the main proof technique when you have infinitely many statements to prove indexed by the integers, such as

1-ring Towers of Hanoi can be won. 2-ring Towers of Hanoi can be won. . . . 487-ring Towers of Hanoi can be won. . . . Every budding mathematician needs to know mathematical induction, and it's a great proof technique to learn first, because it has a standard template (unlike most proof techniques) and yet leaves room for an infinite amount of variety and ingenuity. Thus this course is one of MathPath's foundation courses. But don’t take my word for it. Here is what a MathPath student wrote on an AoPS MathPath forum on June 24, 2010: I took induction last year, and I knew induction before the class. But it was very well taught, and I learned how to write proofs by induction, which was very valuable on the USAJMO. I'd recommend this class to anyone who wants to learn about induction, whether you know it at the beginning or not.

The point: Even if you have done induction before, you don’t really know induction, because it can be used in so many ways in so many parts of mathematics. Every mathematician will probably do 1000 inductions in his/her life. Get 50 under your belt in this course. 2 stars AMC 8, Ms O’Neill Working on problems from old AMC-8 exams, we will study the many ways in which the traditional topics covered in middle school math classes can provide the basis for more challenging questions. Students will become familiar with many problem-solving techniques as well as interesting applications of familiar concepts. This course we be especially helpful for young students who have not yet taken any AMC exams and students who did close to perfect on the 2013 AMC8 and what to go for a perfect score in 2014. 1 star. Analytical Geometry, Dr T (George Thomas) Francois Vieta’s introduction of literal symbols for representing a class of numerical values was followed in 1637 by Rene Descartes’ application of it in Geometry to represent a point and for the representation of lines and curves by equations. He set up an oblique coordinate system and then the rectangular coordinate system. A first degree equation in x and y gives a straight line in this system, and conversely, a straight line has a first degree equation. The discussion goes over very briefly over the various forms of the equation of a straight line, and an example of how algebra is used to “analyze” geometric construction problems. The area of a triangle is obtained in two ways: In terms of the coordinates of the vertices and in terms of the equations of the three lines constituting the sides. Equations of the various conics are obtained from the Homogeneous General Equation of the Second Degree. The method of the rotation of axes is used to determine the conic an equation of the second degree represents. It is shown that given any five points in the plane there is unique conic passing through them. Equations of tangents, normals, subtangents,a

nd subnormals are obtained for the circle first and then for the other conics. Poles and Polars, and the “correlation” between pole and its polar for a conic are shown. The general second degree equation in two variables is shown with its nine affine forms of which ellipse, parabola, and hyperbola are just three. The method of transformation of coordinates to convert a second degree equation to its most simplified form is illustrated. The simplified form is related – “affinis” – to the original and this leads in to a discussion of affine transformations and affine geometry. Whereas as an ellipse is transformed in this manner to another ellipse and not, for instance, a hyperbola, students are shown that there is a transformation, more general, that takes any conic to any other conic. Students are shown such a (projective) transformation – all conics are circles when viewed from the vertex of an infinite cone. This example is used to point out the role played by Analytic Geometry in the mathematical theory of perspective for curves: when a curve is specified an equation, the equation of the perspective view is obtainable by suitably transforming x and y. Pre-requisites: Algebra 2 and Trigonometry, Difficulty – 3 stars

Week 1, Afternoon National MATHCOUNTS, Prof C We will study topics from number theory, algebra, geometry, logic, and combinatorics as well as learn problem-solving techniques by working on problems from old MATHCOUNTS tests. The intention is to limit ourselves to problems from old national MATHCOUNTS tests, not chapter and state. If you feel that chapter and state would be a better place for you to start, consider the AMC-8 breakout in the morning, which has problems comparable to MATHCOUNTS chapter and state. Most days will include a brief Countdown Round practice. Prior contest math experience will be helpful. 2 stars Error-correcting codes, David Clark (Dr C) If you've ever heard bad static in a phone call or seen strange blocks appear on your TV, then you needed an error-correcting code! A code is not a secret -- at least not the kind of code that we'll be playing with. (For that sort of coding, take the Cryptology course.) Our codes are exactly the opposite: A way of writing down a message which makes it easier to understand and protects it from mistakes or errors. In this breakout, we'll transmit messages to each other and even have the chance to play the part of static: the evil corrupter of messages. We'll learn about vectors, matrices, and how to use them to help cell phones, computers, and even deep-space probes to overcome mistakes and make your messages understood. Number theory I recommended as a pre-requisite but not required. 2 stars ! GSP (Geometer’s SketchPad), Ms O’Neill In this breakout, students will learn to use Geometer’s Sketchpad to construct a large variety of sketches. The course assumes no prior experience with GSP, but will move quickly from basic sketches to more advanced ones so that students become familiar with the many tools (how to label, measure, display, make a tool, etc.) which are part of the program. We will start by

learning how GSP mimics traditional construction using straightedge and compass and then learn to animate and transform figures. In addition to using the program to illustrate and explore ideas from Euclidean geometry, we will investigate concepts from algebra, trigonometry, and calculus. On the way, we will draw many beautiful and interesting sketches including spirals (Fibonacci and Theodorus), fractals, and tessellations. 2 stars The Shape of Space, Jon Rogness This breakout will explore ideas in the area of math known as topology, where a donut (torus) and coffee cup are equivalent objects. It will provide a nice followup for any students who have done the previous breakouts this year which covered the torus, although those courses are not prerequisites for this one. We'll start by learning a few basic rules in topology. Then we'll look at the torus, either as a review or brief introduction depending on whether students were in those previous breakouts. Then we'll look at how to construct other two dimensional surfaces, like the Mobius strip and Klein Bottle. As it turns out, in one of the triumphs of mathematics, we can describe how to build all of the so-called "closed surfaces" by sewing together spheres, Moebius strips and donuts. After we deal with surfaces, we'll have the necessary skills to move on to three dimensions. That means we can think about different possible shapes for the universe -- i.e. the Shape of Space -- and explain why the picture below with the dodecahedra might be a model of how our universe is put together. There won't be much in the way of computations or algebra in this breakout, but you should like thinking about three dimensional shapes. (For example, if you like looking at nets and figuring out what the resulting shape is, we'll be doing more complicated versions of that process.) 2 stars Heavenly Mathematics, Glen Van Brummelen How were the ancient astronomers able to find their way around the heavens without anything even as sophisticated as a telescope? With some clever observations and a little math, it’s amazing how much you can infer. Following the footsteps of the ancient Greeks, we will eventually determine the distance from the Earth to the Moon…using only our brains and a meter stick. Along the way, we will develop the fundamentals of the subject the Greeks invented for this purpose, now a part of the school mathematics curriculum: trigonometry. Scientific calculators are encouraged. Marden's Marvelous Theorem, Dr V (Sam Vandervelde) A cubic polynomial with complex coefficients has three roots, which may be plotted in the complex plane. The derivative of this polynomial is a quadratic with two roots, which may also be plotted in the same plane. We will explore the beautiful relationship among these points, involving ellipses, cube roots of unity, and Cardano's formula for the solutions to a cubic equation. 4 stars !

Week 2, Morning

Elementary Graph Theory, Dr B (Jane Butterfield) Most people think of a "graph" as a visual representation of data - a function, assorted information, etc. In the study of graph theory, we define a "graph" to be a set of points, called "vertices", and a set of lines connecting two of those "vertices", called "edges". Graphs of this sort can be used to diagram, understand, and solve many mathematical problems; some of which may surprise you! In this breakout, we will be investigating the fundamentals of graph theory, and discovering problems that can be solved by being diagrammed as graphs. 1 star

Mathematica II, Prof C This course will expand on the topics covered in Mathematica I. We will learn how to define our own functions, how to write loops, how to create animations using the Manipulate command, and how to generate 3D graphics. The exercises will be more challenging than the ones in Mathematica I. 2-3 stars Cryptology, Dr C (David Clark) Repeat of Week 1 course; see it’s description earlier Combinatorics (Basic Counting), Mr M For students who know a little about how to count, but want to know more and get better. We start with the basics: sum rule, product rule, permutations, combinations, the binomial theorem. Then we learn about “combinatorial arguments”, sometimes called proof by story. Time permitting, we will look at inclusion-exclusion, and pigeon-hole arguments. This course is a good prerequisite for various later courses that involve counting, such as The Poisson Distribution, Recurrences, and Proofs you Can Count On (which is about proof by story). 2 stars AMC 10, Ms O’Neill Working on problems from old AMC-10 exams, we will study the many ways in which the traditional topics covered in early high school math classes can provide the basis for more challenging questions. Students will become familiar with many problem-solving techniques as well as interesting applications of familiar concepts. 2 stars Hyperbolic Geometry, Dr T

This is an ideal course for the future mathematician in that it combines history and the axiomatic method. The course begins with a discussion of Neutral Geometry – axioms, the Exterior Angle Theorem, Alternate Interior Angle theorem and the Saccheri-Legendre Theorem. Next: Euclid’s

Fifth postulate (“Fifth”) is added; its equivalence with the Euclidean Parallel Postulate and the Converse of the Alternate Interior Angle Theorem is shown. The efforts of two millennia to prove the Fifth from the Neutral Geometry postulates and the introduction of the hyperbolic postulate. H. Liebmann’s proof: The area of a singly asymptotic triangle is finite. The finiteness of triangular area and Gauss’s proof of the area of a triangle in terms of angular defect. The Bolya formula connecting the angle that a perpendicular of given length to a given line makes with a parallel to the given line is derived. Bolya’s construction of ultraparallels. The Klein, Poincare and Beltrami models.!Pre-requisites: Euclidean Geometry, Trigonometry Difficulty – 3 stars Number Theory I, Dr V Number Theory is the study of the positive integers. How challenging can it be to understand such simple numbers? The wonderful answer is: plenty. The objects are so simple, yet the mathematics is deep. The main topics for NT1 are: Divisibility GCD/LCM, relatively prime integers, the Euclidean algorithm, primes and composites, and an introduction to modular arithmetic. We will prove that there are infinitely many primes, and that every positive integer > 1 has a prime factorization, though the proof that this factorization is unique awaits a later course. If you have seen most of these topics, you are probably ready to begin in NT2. 1 star

Week 2, Afternoon Moving math, Dr B Have you ever flipped a mattress? What about a square mattress, or a hexagonal mattress? Have you ever stood in line? In alphabetical order by last name? How much shuffling did you have to do to get into order? In this breakout session, we will take the expression "doing math" to a whole new level, by exploring certain algebraic structures in a physical way. We will learn about dihedral groups, permutation groups, and of course what a "group" is. Other possible topics include quaternions (and maybe even octonians), polyhedral groups, and the free group. 1 star, no prerequisites Origami, Prof C Most of us are familiar with origami, the traditional Japanese art of paper folding, but you might be surprised to know that there are strong connections between origami and mathematics. During this breakout session we'll explore a number of mathematical ideas with folding projects. In some cases we'll create geometric objects (such as dodecahedra) which illustrate mathematical ideas. In other cases, paper folding can be used to solve a mathematical problem. If you like folding animals and other objects out of origami, this could be a fun and interesting breakout for you. However, previous experience or skill with origami is not required to take this breakout – just a willingness to learn and attention to detail in your folding. 1 star

Information theory, Dr C If u cn rd ths, u r wlcme n ths brkout! If you've ever sent a text, spoken really fast, or attached a file to an email, then you've used compression. Compression involves throwing away information in order to make it shorter and smaller, which runs the risk of making it hard to understand. The first sentence shows how the English language is filled with extra letters which helps us make sense of even highly compressed messages. In this breakout, we'll discover how far can you shorten a sentence before it loses its meaning. We'll see several ways to do this, including the secret behind ZIP files. Finally, we'll play a famous game which lets us measure how much meaning every word in the English language has. 3 stars Kenken extended to Kendoku, Mr L Take the basic KenKen grid and add the Sudoku constraint of the Sudoku boxes. Kendoku puzzles can only be 6X6 or 9X9. That is because of the addition of 6 Sudoku boxes each 2x3 in the 6X6 puzzle and the standard 9 Sudoku boxes each 3x3 in the 9X9 puzzle. Week 1 is not a prerequisite but I recommend trying some simpler KenKen puzzles over the weekend prior to class. All of the techniques used in Week one will be taught again using different examples. Deductive reasoning, extensive use of logic AND the ability to persevere in the face of frustration are the only prerequisites for this course. Guessing will not be tolerated. 2 stars Spherical Trigonometry, Glen To find your way through the heavens, along the earth, or across the oceans, you need mathematics. But the math you learn in school mostly takes place on a flat surface --- not the sphere of the heavens, or the earth. To navigate properly, we develop a completely new trigonometry that allows us to find our way around a sphere. We shall discover surprising symmetries and a rich world of theorems, many of which are beautiful and unexpected extensions of some of the most familiar geometric theorems we have learned in school. Scientific calculators are encouraged. 3 stars Argentine Area , Dr V We all know that the area of a triangle equals one-half base times height. But there are far more exotic formulas for the areas of triangles and quadrilaterals, involving quantities such as the inradius of a triangle or the cosine of the average of a pair of opposite angles in a quadrilateral. We'll cover lots of geometry in our quest to master an assortment of area formulas. 3 stars !

Week 3, Morning

Note: Returnees, in addition to taking one morning and one afternoon breakout, will generally attend a pullout history section with Prof B (Owen Byer) while Coach D is giving his history plenaries during Weeks 3 and 4, pretty much the same plenaries he gave last year. The pullout will generally do various topics with an eye towards how they were developed historically. Graph Ramsey games, Dr B It would be surprising if there were ten students in my class who had the same birthday. It would not be surprising if there were ten students in Massachusetts who had the same birthday. That is because when things are large we expect patterns to show up. Ramsey Theory answers the question "how big does the group have to be for a particular pattern to always happen?" In the context of graph theory, we might ask how many edges a graph can have before it has to contain a clique of size three. Questions like this form the basis of Graph Ramsey theory. In this breakout session, we will answer some Graph Ramsey questions, and learn just how hard the unanswered ones are. Then we will spend the rest of our time playing games -- I mean, "considering Graph Ramsey Theory from a competitive perspective." Some of theses games will be between two intelligent players, and some will be between an intelligent player and a random adversary. We will show how these games are related, and try to decide who wins them. This area is still very recent, and many easy-to-ask questions are still unanswered (sand are possibly very difficult). 1-2 stars, nice if you know some graph theory but not necessary

Number Theory II, Prof D (Matt DeLong) Suppose a fruit stand owner wants to arrange oranges neatly. If the oranges are arranged in rows of 5 then there are 2 left over, if arranged in rows of 6 then there is 1 left over, and if arranged in rows of 7 there are 3 left over. How many oranges could the owner have? Is there a unique solution? Answering this question requires a deeper understanding of modular arithmetic than we achieve in NT1. So in NT2 we discuss inverses mod m, the Chinese Remainder Theorem, Fermat's Little Theorem, Wilson's Theorem, the Euler phi-function and Euler's extension of Fermat's Little Theorem. These results will allow us to answer other questions as well, such as: How many ways can 1 be factored? What happens when one repeatedly multiplies a number by itself? And because the answers to such questions are so interesting, we can all enjoy quick divisibility tricks, and other applications.

Elementary Logic, Coach D If you attend MathPath, you cannot be a cow. Bessie is a cow. Most of us would correctly deduce that Bessie cannot attend MathPath, but how do we know this, and how can we prove it? Logic serves as the foundation for our reasoning, setting out formally the rules we understand intuitively. Much as algebra allows us to generalize relationships among numbers by using variables to represent quantities, we will use symbols to represent propositions, demonstrating the

form or structure of an argument (premises leading to a conclusion). Then we will formally prove these arguments using rules such as Modus Ponendo Ponens, Modus Tollendo Tollens, and Reductio Ad Absurdum. We will explore truth tables, semantic trees, symbolic logic proofs, and potentially predicate logic. Also, there are paradoxes which seem to defy common sense. We shall see how putting some paradoxical examples into the regimented forms demanded by logic causes the paradoxical element to disappear. 1-2 stars

AMC 12, Mr M

This course is for students planning to take the AMC 12 for the first time next spring, and for students who have taken it before but want to improve their score, perhaps enough to qualify for the AIME or an Olympiad. We will emphasize topics covered on the AMC 12 that are not covered on the AMC 10, namely, precalculus topics, such as functions, trigonometry, complex numbers, polynomials, and solid geometry. Some days we will do problems all on one or two topics, some days we will take a test under test conditions, even though we won’t have a full 75 minutes. Prior contest math experience recommended. 2-3 stars

Topics in Advanced Graph Theory, Kip We will study Ramsey Numbers, from the basic idea through to an exploration of how the search continues for larger ones, and we’ll prove an important theorem that helps narrow that search. There will be colouring tasks which underpin a greater understanding of the concept. We will also study two theorems associated with Hamiltonian Cycles in graphs, one providing a sufficient condition and the other a necessary condition for the existence of a HC in a graph. Time permitting, we will also look at algorithms associated with the Chinese Postman Problem, and the achieving of upper and lower bounds for the Travelling Salesman Problem. 2-3 stars. Prerequisite: some previous experience with graph theory, such as from taking the Elementary Graph Theory Breakout. Boxing Dots, Dr V Given a square array of dots, how many can we enclose using a square with vertices from the array? This innocent question will lead us on a merry chase that will encompass important ideas from both geometry and number theory, and provide plenty of opportunity for conjecture and proof. 2-3 stars

Week 3, Afternoon

The Four Color Theorem, Dr B There are many reasons to color a map; you might want to record demographic information about a region, you might want to indicate political leanings, or you might just want to make the boundaries between regions easier to see. The sort of coloring we will consider in this session is a "proper" coloring, which means that no two adjacent regions have the same color. In the 19th

century, mathematicians started to wonder how many colors they needed in order to "properly" color a map. A student said it was pretty obvious that four colors were always enough -- what followed was over 100 years of argument, partial proofs, and April Fools jokes as mathematicians tried to prove (or disprove) this claim. In 1976, it was finally announced that the Four Color Theorem had been proven; we will discuss why some mathematicians are still unhappy with this proof, and what further progress has been made since then. In this session, we will follow the history (and drama) of the now-famous Four Color Theorem. We will prove the Six Color Theorem and the Five Color Theorem, and explore some of the false proofs of and false counterexamples to the Four Color Theorem. The focus will be on this history of the problem; no previous exposure to map coloring is necessary, and students who are already familiar with the Four Color Theorem will probably still learn something new about the history of the problem from this session. 1-2 stars, no prerequisites. This will be taught from a historical perspective more than a graph theory perspective, so that it will be accessible to any student. Poisson Distribution, Prof B The Poisson Distribution is used to find the probabilities of the number of occurrences of some event in a fixed interval of space or time. Typical questions include the following.

1. What is the probability that there are two typographical errors on a random page of a given novel?

2. What is the probability that three different calls are made to an Emergency Response Unit in a given hour?

We will derive the formula for the Poisson probability distribution from beginning principles, discovering the binomial distribution and the number e along the way. We will examine many applications to real world phenomena. Students taking this course should be familiar with basic probability and familiarity with combinations and permutations will be very helpful. 2-3 Stars Knot Theory, Prof D Knots have fascinated farmers, sailors, artists, and myth-makers since prehistoric times, both for their practical usefulness and for their aesthetic symbolism. For over 200 years, mathematicians have intensely studied knots, both for their own intrinsic mathematical interest and for their potential application to chemistry, physics and biology. Today, Knot Theory is an active field of mathematical research with many important applications. It is visual, computational and hands-on, and there are many easily stated open problems. This class will give an introduction to the fundamental questions of Knot Theory. It will take students from simple activities with strings to open problems in the field. It will answer such questions as, “what is the difference between a knot in the mathematical sense and a knot in the everyday sense?,” “how do I tell two knots apart?,” “how can I tell whether a knot can be untangled?,” and “how many different knots are there?” 2-stars Games People Play with a KenKen base, Mr L In addition to a mixture of Kendoku problems (the difficult 6X6 is my favorite), we will try more complex (I call them exotic) KenKen problems which can be formed using the principles described above in Week 1 by:

1) omitting the symbols +, !, " and ÷, thus leaving them as yet another unknown to be determined and/or 2) allowing 0 (so the numbers 0 to 8 are used in a 9X9 puzzle) or 3) Allowing the Mod operator as well as the basic 4 operators +, -, / and X and.... 4) We may try a fractional KenKen puzzle (two numbers in some boxes). We will do 6X6 Kendoku and some of the exotic puzzles described in 1) to 4) above. I will teach solving tricks to start them using several 9X9 puzzles and then let you complete the puzzle in your free time if you wish. Weeks 1 or 2 are not prerequisites but you have to show basic knowledge of solving a KenKen puzzle to be allowed in this class. All of the techniques used in Week one will be taught again using different examples. Deductive reasoning, extensive use of logic AND the ability to persevere in the face of frustration are the only prerequisites for this course. Guessing will not be tolerated. 3 stars Elliptic Geometry, Dr T In this course we explain why there are exactly three 3-dimensional constant-curvature geometries – Euclidean, hyperbolic and elliptic. Whereas the Euclidean and hyperbolic geometries are neutral geometries, we show that Elliptic Geometry is not a Neutral Geometry. However, Elliptic Geometry complements the Euclidean and Hyperbolic geometries in terms of the number of parallels, namely none. We discuss Elliptic Geometry using two models – Spherical Geometry and the Klein Circle model – to show also that it is a projective space with an elliptic metric. We also prove some fundamental properties of a sphere including that a geodesic is a great circle on the sphere. Pre-requisite: Trigonometry. Those who took Spherical Geometry will find some overlap due to the use of the sphere in building the model of elliptic Geometry. 3 stars Taming the Torus , Dr V. A torus is the mathematical term for a shape such as a donut or bagel. We will examine the geometry of the torus, followed by a discussion of its topological properties, including triangulation, graphs, and games on the surface of a torus. 1 star !

Week 4, Morning

Number Theory III, Prof Davidoff MathPath campers are all familiar with the integers, in particular, those we call primes, and their special role inside the set of rational numbers (quotients of integers), denoted . For example, we know that integers can be factored uniquely into products of prime powers. We will look at a larger set of numbers, namely the set , where . You may know that we can add, subtract, multiply, and divide such numbers, so they have a very rich arithmetic, just as the rationals do. We will find out what the “integers” and “primes” are in this larger set, and decide if each of these new “integers” can be uniquely factored into products of our new “primes”. The key to studying the arithmetic of is to understand all solutions (x,

y, p) to the equation , where x and y must be ordinary integers and p is an ordinary prime number, so we will begin with this. Prerequisite: Knowledge of the material in NT II. 3-4 stars Infinity, Coach D There’s a great deal to say about infinity, and some of it has been said by authors from Norton Juster (The Phantom Tollbooth) to Douglas Adams (The Hitch-Hiker’s Guide to the Galaxy). Philosophers continue to argue over whether there are contradictions at the heart of the idea of infinity. In the meantime, mathematicians have been able to proceed to work with and around infinity. The breakouts will start with getting everyone to say something about infinity (but there’s only a finite amount of time to say it). We shall compare our views of the subject with what has been said in the past. We shall talk about Zeno’s paradoxes and the problems that arise from treating infinity as a number. After all, if we can talk about a circle as a polygon with an infinite number of sides, how does that help us to understand either circles or polygons? One of the best ways to deal with infinity is to translate sentences that talk about ‘infinity’ into sentences from which the word has been removed. We can talk about an ‘infinity’ of primes, but we don’t need the idea of infinity. There are plenty of places where infinity seems to show up where a good translation avoids some of the problems that the idea creates. In fact, we’ll even be able to show how having ‘infinity’ in a problem makes it easier to solve, whether it involves repeating decimals, geometric series, continued fractions, or nested radicals. One of the real accomplishments of the nineteenth century was figuring out how to talk about calculus without involving the notion of ‘infinite’ or ‘infintely small’. If we want to allow the notion of infinity into our mathematics, we can follow the ideas of Georg Cantor, who built up a whole hierarchy of infinities. We can come up with (not surprisingly) paradoxical results about how big the sets of whole numbers, rational numbers, and algebraic numbers are. Then we can take a jump to the real numbers (and perhaps to Professor Conway’s ‘surreal numbers’). One of the issues that mathematicians continue to debate is the relationship between the number of whole numbers and the number of reals. Can we draw a picture of infinity? After all, artists might want to draw a picture of railroad tracks, which could be a challenge if the canvas isn’t infinitely big. The invention of perspective

had an influence on projective geometry, and the influence of mathematics in art has been equally conspicuous. The Dutch artist M.C. Escher was not formally trained in mathematics, but some of his later work was inspired by mathematicians and their work in hyperbolic geometry. We shall end by tying Escher’s art work into music and even logic. 2 stars !Induction,"Mr"M""""""Repeat"of"Week"1"course"""Number Theory II, Dr W (William Meyerson) Repeat of course from Week 3, probably with some different problem sets; see description there. Integer Partitions sVsen-san (Tom Roby) How many ways can one you write 4 as a sum of positive integers? If order matters, so that we consider 3+1 and 1+3 to be different, then the answer (8) is an easy application of very basic combinatorics. But if we only care about the (unordered) *set* of addends, the answer is 5. This simple question leads to a rich, beautiful, and surprisingly challenging mathematical area on the intersection of combinatorics and number theory. We will scratch a few of its many surfaces, including some beautiful bijections, clever counting, revealing recursions, delicious diagram dissections, and playful polynomials. For this breakout, some previous experience with elementary combinatorics (sum rule, product rule, permutations, combinations, the binomial theorem) and facility with manipulation of polynomials and geometric series will be helpful. 3-4 stars "" Discovery Learning, Maria This course is based on a Hungarian style method of teaching, called the “Posa Method”, which focuses on the discovery of mathematics. That is, the students not only have to prove the results, but also they have to discover what to prove. The instructor will at most give occasional hints. The philosophy of this method is best described by the following quote: "mathematics is not the private affair of a selected few, but the fun of discovering and autonomous thinking is everyone's right." Participating in discovery learning involves building on one’s previous knowledge and developing new ideas to find creative solutions. Thus, in this course we will encounter a variety of mathematical problems and techniques to solve them. The choice of mathematical areas we will cover will depend on student preference and background, but my focus will be proof of

impossibility, basic counting, but more generally, creative problem thinking. The problems given will be in the form of individual work, team play, quick questions, etc. This course is suitable for those who are persistent with their mathematics. Prerequisites: number theory, basic counting. 2 stars "Projective Geometry, Dr T

Euclidean Geometry is the geometry of lines and circles: its tools are the straight edge (unmarked ruler) and compass. There is a geometry where its constructions need only a straight edge. In this geometry a straight line joins two points and two lines never fail to meet. It turns out to be simpler than Euclid’s but not too simple to be interesting. This is Projective Geometry which turns out to be the mother of all geometries, for Euclidean, Elliptic, Hyperbolic and Minkowskian Geometries are but special cases of it. The course begins with Pappus’s ancient theorem. The discussion then proceeds as follows: The new Geometry’s axioms, the principle of duality, the concept of projectivity, Harmonic sets, The Fundamental Theorem of Projective Geometry, and Desargue’s Theorem. Pre-requisites: None Difficulty – 3 stars !

Week 4, Afternoon " Inversions, Prof B An inversion of the plane is a transformation of the points in the plane that preserves various properties of lines, circles, and the way they intersect each other. Inversions are a tool that can be used to provide elegant proofs of many geometric facts. They can be used to create interesting and artistic distortions of figures. Furthermore, properties of inversions can be combined with software like Geometer's Sketchpad to create dynamic figures (for example, Steiner's Porism), and we will explore these interactions. Students taking this course should have some experience in proving theorems from geometry, though no specific knowledge is necessary. 3 stars

"Mr L

National MATHCOUNTS, We will work on a selection of old national MATHCOUNTS problems. Each day we will work on 3 or 4 problems that will perhaps illuminate the problems in the set you will work on that day. You will do a complete Sprint set from a prior year...and not a recent year. We will then explain the solutions to selected (your choice) problems. If the class votes to do Target sets we can have a day of those as well. 2 stars

""

Inequalities,!Dr"W""Often"questions"in"Olympiad;type"mathematics"competitions"require"you"to"compare"two"expressions"and"show"that"one"is"larger"than"the"other."Inequalities"are"the"standard"tools"that"enable"one"to"do"so.""Starting"with"the""trivial""inequality"x2">"0"whenever"x"is"real"and"nonzero,"we"will"discuss"both"the"AM;GM"inequality"and"the"Cauchy;Schwarz"inequality"along"with"various"ways"these"can"be"used."3"stars"""Proofs You Can Count On, sVen-san Many equalities among, say, binomial coefficients or Fibonacci numbers that are commonly proven by induction or symbol pushing have beautiful bijection proofs. These often give additional insight into why these identities hold, working at a conceptual rather than a manipulative level; once you see such a proof, you are usually convinced that there could not be a better way to prove that theorem. The famous mathematician Paul Erdös used to say they were proofs from “The Book”.

For example, why does the basic recurrence in Pascal’s triangle hold? We think of nk⎛

⎝⎜

⎞

⎠⎟ as

counting the number of k-element subsets of [n] : ={1,2,…,n}. Given a k-element subset S of [n+1], either it contains n+1, in which case S – {n+1} is a (uniquely determined) (k-1)-element subset of [n], or it does not contain n+1, in which case S is already a k-element subset of n. These mutually exclusive cases show that

n +1k

⎛⎝⎜

⎞⎠⎟=

nk −1

⎛⎝⎜

⎞⎠⎟+nk

⎛⎝⎜

⎞⎠⎟

In addition to examples among and between binomial coefficients and Fibonacci numbers, we will also consider Catalan numbers. 3 stars " Advanced Mandelbrot, Dr V The Mandelbrot Competition is a contest that has been offered each year to high school students across the United States. We will examine questions from the more difficult level, implementing a variety of formats. This breakout will be useful for students who wish to hone their contest problem solving skills and learn advanced techniques for answering the sorts of questions that typically appear on Mandelbrot or the AIME. 3-4 stars Elementary Graph Theory, April (April Verser) Most people think of a "graph" as a visual representation of data - a function, assorted information, etc. In the study of graph theory, we define a "graph" to be a set of points, called "vertices", and a set of lines connecting two of those "vertices", called "edges". Graphs of this sort can be used to diagram, understand, and solve many mathematical problems; some of which may surprise you! In this breakout, we will be investigating the fundamentals of graph theory, and discovering problems that can be solved by being diagrammed as graphs. This version of the course will cover the same standard ideas as Dr B’s version in Week 2, but the examples and problems may be 50% different. 1 star

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