MathPath Breakout Catalog 2016

V4.0 7/15/16
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

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 4x4 grid, 1 through 5 for a 5x5, etc. so that no digit appears more than once
in any row or any column. Grids range in size from 3x3 to 9x9. 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. 1-2 stars depending on what the students want in the way of
difficulty.

Proof by Story, Ms ONeill (Granya ONeill)

Many identities involving, say, binomial coefficients or Fibonacci numbers that are commonly
proven by induction or manipulating algebraic expressions also have beautiful bijection
proofs. Such bijection proofs are often called proof by story, since we come up with two different
ways (stories) to count the same thing. These proofs often give insight into why the identity
holds, and are so elegant that you will be convinced that there could not be a better proof of that
particular theorem. Paul Erdos said that such proofs were from The Book. In this breakout,
we will devise as many proofs by story for identities involving binomial coefficients and
Fibonacci numbers as we can. 2-3 stars

Cryptology, Prof Rogness (Jon Rogness)

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

Number Theory I, Dr T (George Thomas)

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

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. 1 star

Week 1, Afternoon

Mathematical Origami, Prof C (Silva Chang)

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) to illustrate mathematical ideas. In
other cases, paper folding will be used to solve a mathematical problem. If you like folding
origami models, 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
AMC 10, Ms ONeill (Granya ONeill)
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

The Shape of Space, Prof Rogness (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

Analytical Geometry, Dr T (George Thomas)

Francois Vietas 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 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 a unique conic passing through them. Equations of tangents, normals, subtangents,
and 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 an ellipse is transformed in this manner to another ellipse and not, for
instance, a hyperbola, there is a transformation, more general, that takes any conic to any other
conic. It is the 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. 3 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, its
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. 2 stars

Taming the Torus, Dr V (Sam Vandervelde)

Torus is to bagel as sphere is to cantaloupe. The torus presents a splendid variety of mathematical
aspects for our examination. We will discover the geometric formulas associated with a torus,
catalog the topological properties of a torus, investigate graph embeddings on a torus, even
explore the possibilities for art and puzzles with a torus. 1 star

Week 2, Morning

Mathematica II, Prof C (Silva Chang)

This course will expand on the topics covered in Mathematica I. We will learn how to define our
own functions, 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

Basic Counting (Combinatorics), Dr C (David Clark)

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. 2 stars

Number Theory II, Prof D (Matthew 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.
2 stars

Induction, Ms ONeill (Granya ONeill)

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 dont 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 dont 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

Hyperbolic Geometry, Dr T (George Thomas)

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: Euclids
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. Liebmanns proof: The area of a singly asymptotic
triangle is finite. The finiteness of triangular area and Gausss 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. Bolyas construction of
ultraparallels. The Klein, Poincare and Beltrami models.!Pre-requisites: Euclidean Geometry,
Trigonometry. 3 stars

The Fabulous Fibonacci Array, Dr V (Sam Vandervelde)

All of us are familiar with the sequence of Fibonacci numbers. Many of us are cognizant of other
generalized Fibonacci sequences, such as the Lucas numbers. Some of us are familiar with
identities common to all the generalized Fibonacci sequences. By the end all of us will have seen
the "right way" to fit all these sequences together into a single two-dimensional array having a
plethora of astounding properties. 2-3 stars

Week 2, Afternoon

National MATHCOUNTS, Prof C (Silva Chang)

We will learn problem-solving techniques by working on problems from old MATHCOUNTS
tests, covering topics from number theory, algebra, geometry, logic, and combinatorics. The
intention is to limit ourselves to problems from old national MATHCOUNTS tests, not chapter
and state. Most days will include a brief Countdown Round practice. Prior contest math
experience will be helpful. 2 stars

Error-Correcting Codes, Dr C (David Clark)

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

Elementary Logic, Prof D (Matthew DeLong)

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

Geogebra, Ms ONeill (Granya ONeill)

Geogebra is an interactive geometry, algebra, statistics, and calculus program developed by
mathematicians and engineers for teaching and learning mathematics. In this breakout, students
will learn to use Geogebra to construct a wide variety of sketches. The course assumes no prior
experience with Geogebra but will move quickly from basic models to more advanced ones as
students become familiar with the many tools which are part of the software. We will start by
learning how Geogebra mimics traditional construction using straightedge and compass and then
consider applications in analytic geometry. In addition to illustrating and exploring ideas from
Euclidean geometry, however, we will also investigate concepts from algebra, trigonometry, and
calculus. Our emphasis will be on applications not easily performed on a graphing
calculator. During the week, we will draw many beautiful and interesting sketches including
spirals (Fibonacci and Theodorus), fractals, and tessellations. 2 stars

Spherical Trigonometry, Glen Van Brummelen

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

Ford Circles and Farey Tales, Dr V (Sam Vandervelde)

Beginning with a pair of parallel lines, we draw circles tangent to everything in sight. Quite
predictably, we will develop useful geometric techniques and employ a bit of algebra in the
process. Rather unexpectedly, a nice piece of number theory materializes along the way.
Somewhat disappointingly, no wild trouses show up, but a wild value of the Riemann zeta
function does make an appearance towards the end. 3-4 stars

Week 3, Morning

Guessing Games, Dr C (David Clark)

I'm thinking of a secret whole number from 1 to 10. How many yes-or-no questions do you need
to ask in order to figure it out? What if it's 1 to 1,000,000? How about 1 to n -- and how do you
know that you couldn't use fewer questions? In this breakout, we will start with those problems
and then go much farther. For example: What if I'm allowed to lie -- but only once? Twice?
Prerequisites include a familiarity with simple proofs and Basic Counting (Combinatorics). It will
help to have taken Error-Correcting Codes, but it is not required. Come prepared to play! 2 stars

Infinity, Prof D (Matthew DeLong)

The concept of infinity has long fascinated mathematicians, philosophers, theologians, and artists.
The paradox and power of the infinite inspires wonder and fascination. In this breakout we will
wrestle with some of the paradoxes that arise when working with the infinite, as well as see the
power that results from incorporating the infinite into our mathematics. We will follow the ideas
of Georg Cantor, whose creative ideas systematized the study of the infinite in a way that was
profoundly brilliant and (at the time) controversial. We will see how Cantors ideas allow us to
understand the nature of the real numbers more deeply, as well as to answer questions like, Are
there different sizes of infinity? and Is there a largest infinite set? We will also relate the
infinite to different areas of mathematics as well as to art, music, and literature. 2 stars

AMC 12, Dr K (Sinan Kanbir)

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 to illustrate one or
two concepts or problem-solving techniques; some days we will take a test under test
conditions. Prior contest math experience recommended 2-3 stars

Integer Partitions, sVen-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

Cryptology, Prof Rogness (Jon Rogness)

See the description in Week 1 morning.

Elliptic Geometry, Dr T (George Thomas)

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

Week 3, Afternoon
Information Theory, Dr C (David Clark)
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. 4 stars

Knot Theory, Prof D (Matthew DeLong)

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 handson, 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

Never Bored with Chess, Coach D (Thomas Drucker)

There are many features connected with the game of chess of interest to a mathematician. We
can start with the board itself. Professor John Watkins (who has taught at MathPath) wrote a
book with the title Across the Board in which he looks at a number of those aspects of the board.
There are questions about geometry (and the length of the paths followed by various pieces) and
about combinatorics and graph theory. In addition, there are variants on the standard chess board
that involve connecting the edges and getting a torus. You may never have to play a game on a
toroidal board but you can learn to appreciate some of the tricky features of its topology.
Then there are questions associated with the game and not just the board. For example, we
can calculate the number of possible positions after a small number of moves (its quite large) as
well as the number of possible chess games (thats even larger). There are problems associated
with positions that involve one player winning the game in a thousand moves or more. How
these numbers are associated with the way the pieces move and how many pieces there are is a
reminder of how quickly certain kinds of numbers can grow.
Finally, there are mathematically related questions about how to program a computer to play
good chess. Chess programs of dubious quality go back to the earliest days of programming
computers, but algorithms strengthening the play of a machine depend on being able to associate
numerical values with positions. We shall talk a little about heuristics and compare chess with
other games (like tic-tac-toe and Go) in this respect, especially since the recent triumph of a Go
program over one of the worlds strongest players. This breakout will not necessarily make you
a stronger chess player but it will give you many new ways of thinking about the game. In
addition, we shall be looking at positions on the board and figuring out what must have happened
in order for those positions to arise. This sort of analysis brings up questions that arise in various
branches of mathematics. 1-2 stars

AIME Problems, Dr K (Sinan Kanbir)

Practice with lots of AIME problems. For students who have taken the AIME but not done well

yet, and students who hope to qualify next year. 3 stars

Number Theory II, sVen-san (Tom Roby)

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.
2 stars

Discrete and Computational Geometry, Prof Rogness (Jon Rogness)

If you build an art gallery, how many people do you need to guard it? This is a geometric
problem, but it has a very different flavor than the geometry questions youve seen in other
geometry courses, and we need different tools to solve it protractors, rulers, or area formulas
wont be much help here; the solution turns out to depend on a counting argument based on the
number of walls, and not the specific shape of the gallery!
During this breakout well learn more about the art gallery problem, and also explore related
problems which fall under the umbrella of discrete and computational geometry. Well learn
new ways to compute distances between points and why wed care! Once we have those new
notions of distance, well study the shortest way to connect a set of points in the pane. The
computational in the title means that well do some computations based on the geometric
objects were working with; it doesnt mean well be using computers, although many of the
arguments well look at can be implemented with computer programs. In one case, well use
soapy water to do our computations for us! 2 stars
Prerequisites. Familiarity with basic Euclidean geometry: distance formula, polygons, and conic
sections. Some of our solutions will use proof by induction, so it would help to be familiar with
induction.

Week 4, Morning

Special Relativity: The Mathematics of Paradox, Silas Johnson

Einstein's Special Theory of Relativity, at heart, is not about physics so much as the geometry of
the 4-dimensional space-time we live in. Motivated by the paradoxes that plague older ways of
thinking about physics, we'll discover the equations that define this geometry. We'll finish by
using these all-important equations to resolve the paradoxes we explored earlier in the week. If
we have extra time, we might take a look at why faster-than-light travel is impossible. 3 stars
Triangle and Circle Geometry in Math Competitions (AMC10/12, AIME and USAJMO),
Isil Nal
Often geometry questions in competitions such as the AMCs require you to use a variety of
techniques. We will discuss the following topics and solve problems similar to AMC 10/12,
AIME and USA(J)MO levels:

similarity,
collinearity of points and concurrency of lines,
finding area using techniques other than one-half base times height formula,
finding lengths in circles using power of a point, Ptolemys Theorem, and kissing circles.

3-4 stars
Burnsides Lemma: Counting in the Presence of Symmetry, Osman Nal
In how many ways (up to rotations and/or reflections) can the sides of a regular hexagon be
colored using 3 different colors? In how many ways can the faces of a cube be numbered from 1
to 6? In how many essentially different ways can 5 crosses (Xs)and 4 noughts (Os) be arranged
on a tic-tac-toe game board?
All of these types of problems involve the notion of counting in an environment where symmetry
exists. A good understanding of the notation of abstract algebra certainly helps. These sorts of
combinatorics problems involving the symmetry of 2- and 3-dimensional objects will be
addressed in this course with the use of a method called Burnsides Lemma. Along the way we
will learn a little bit of abstract algebra, which is the language of symmetry. 2 stars

Intermediate Combinatorics, sVen-san (Tom Roby)

Taking up where Basic Counting left off, we will look at a rubric called "The Twelvefold
Way" that systematizes many fundamental counting problems. This will naturally lead us to
consider some special counting numbers, including those (mis)named for Bell and Catalan.
Refining the above numbers will give us some of the combinatorial triangles discussed in the
plenaries, including those of Stirling and Narayana numbers, which we will consider in more
depth. 3 stars

Topics in Advanced Graph Theory, Kip Sumner

We will study Ramsey Numbers, from the basic idea through to an exploration of how the search
continues for larger ones, and well 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.

Week 4, Afternoon
Boole-a, Boole-a, Coach D (Thomas Drucker)
Logic has a long history. So does arithmetic. It took until the nineteenth century for someone
(George Boole) to make an association between the two that has proved to be remarkably fruitful.
The connections also extend to set theory and probability. Well draw some of these connections
while laying a foundation for the individual areas.
Logic goes back to Aristotle (at least) and his notion of a syllogism. We shall talk about the idea
of assigning truth values to statements and how the truth values of compound sentences can be
calculated from the truth values of the components. We shall also look at tautologies (statements
that are always true) and contradictions (statements that cannot be true), as well as paradoxes that
may look contradictory but are not quite.
You are all likely to be familiar with arithmetic. That may have required learning a fair number
of calculations like 7 times 8. In this breakout well be dealing with arithmetic modulo 2, so the
only numbers you have to be able to calculate with are 0 and 1. You can think of this as dividing

all numbers into evens and odds. That may not seem to be a fine distinction, but it turns out to be
enough to simplify some hard questions in algebra. It is also connected with logic, where we
tend to use 0 and 1 as the truth values for statements.
There are basic laws of algebra which apply to the manipulation of real numbers. Many of them
apply to the manipulation of sets as well, where the operation of intersection corresponds to
multiplication and union to addition. The correspondence is not exact, however, and we shall
look at some of the differences. The algebra of sets is called a Boolean algebra in recognition of
what George Boole did in the 1840s and 1850s. We shall look at some of the interesting
features of a Boolean algebra such as duality and tie it in with logic. By the end of the breakout,
you should be able to recognize some of the structural similarities between algebra, logic, and set
theory at a fundamental level, and this introduces the idea of what is called an isomorphism. 1-2
stars

Complex Number Theory: Gaussian Integers and Gaussian Primes, Silas Johnson
Ever wondered why number theory only ever seems to think about real numbers? Is 2+3i prime
or composite? How can you tell? Can you take the prime factorization of a complex number? In
this course, we'll extend familiar concepts from number theory, such as divisibility, modular
arithmetic, and prime factorization, into the land of complex numbers. Prerequisite: Some
knowledge of number theory (on the level of Number Theory I); Number Theory II is not required
for this class. 2-3 stars

Kendoku, Mr L (Al Lippert)

Kendoku is a combination of KenKen and Sudoku. 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 4x4 grid, 1 through 5 for a
5x5, etc. so that no digit appears more than once in any row or any column. 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.
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.
* Last year one of the students pointed out that an 8x8 Kendoku puzzle is also possible.
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

Linear Set Geometry, Ryan Matzke

You are familiar with number algebra and perhaps vector algebra. These are examples of linear
algebras, because there is addition and scalar multiplication. In the plane, or any Euclidean space,
there are also linear point and linear set algebras. We can take scalar multiples and sums of sets,
which is quite different from taking their unions. For instance, if we let X and Y be the sets of all
points on the x-axis and y-axis, respectively, then X+Y is the entire Euclidean plane, but the
union of X and Y is just the axes.
Combining the ideas of linear point and set algebras with the traditional set algebra notions of
subsets, unions, and intersections will allow us to treat certain important geometric ideas with
powerful algebraic tools. Geometrically speaking, we discuss convex, balanced, and symmetric
sets, along with the concept of hulls and kernels. For instance the convex hull of S is the smallest
convex superset of S.
As we go through the material, we will prove what we discover, and discuss what is needed for a
good proof. By the end of the course, we will be asking interesting questions about how these
various ideas interact, like: Is the sum of the hulls the hull of the sum?
This course is a compressing of a two-week course, so it will have a bit of a faster pace. In
addition, students should consider that this will be a proof-writing course. Prerequisite: Vector
algebra. Suggested: Some familiarity with proofs. 4 stars
Equations, Functions and Polynomials in Math Competitions (AMC10/12, AIME and
USAJMO), Isil Nal
Equations commonly arise in every branch of mathematics including all the subjects covered in
math competitions such as AMC10/12 and AIME. In this class, we will study and solve AMC
10/12, AIME and USA(J)MO problems on algebraic equations, functional equations, remainder
theorem, polynomials with rational roots, and Vieta`s Formulas. 3 stars
Applications of the Nine-Point Circle, Osman Nal
This course provides a great opportunity to study the nine-point circle and some of its many
applications in triangle geometry. It turns out that (i) the feet of the three altitudes of any triangle,
(ii) the midpoints of its three sides, and (iii) the midpoints of the line segments from the
orthocenter to the three vertices all lie on the same celebrated nine-point circle. This circle has
many exciting properties, such as its center lying on the Euler line, and applications such as the
Feuerbach Theorem. It can also be used in conjunction with other concepts and well-known
theorems to help solve harder geometry problems. Get ready for some geometric transformations
such as dilations and inversions along the way! 3 stars

Projective Geometry, Dr T (George Thomas)

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 Euclids 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 Pappuss ancient theorem. The discussion then proceeds as follows: The
new Geometrys axioms, the principle of duality, the concept of projectivity, Harmonic sets, The
Fundamental Theorem of Projective Geometry, and Desargues Theorem. Pre-requisites: None. 3
stars

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