# Challenging All Campers

## Program Description

Epsilon Camp is for kids who are captivated by math. We provide a rich environment, exposing them to some of the most substantial mathematical ideas of the last two thousand years. We have developed a curriculum that is both accessible and challenging to the brightest of young minds. With professional mathematicians as our faculty, we are able to stretch any part of this material to challenge all campers.

The camp activities are intended to provide engagement in many different ways—both through serious study and serious play. Most days, campers attend three courses. Time is set aside for self-directed learning and fun, including puzzles, math games, and hands-on activities.

Campers are organized into groups of approximately 10 campers per group within the following levels. The number of groups per level will vary from year to year, depending upon enrollment.

**Note: The curriculum changes slightly from year to year. The following are the classes that were taught in 2019.**

## Pythagoras Level, Ages 7–8

We design the academic program for ages 7–8 in a way that requires minimal background, but still introduces campers to a wide variety of deep ideas that they will continue to see throughout their mathematical careers. Topics include:

**Number theory**: prime numbers, divisibility tests, arithmetic in different bases, and more.**Computational mathematics**: recurrences for figurate numbers, Fibonacci numbers, generalizations, and binary information encoding.**Geometry**: polygon angles, plane tilings, polyhedra, and fractals.**Counting**: permutations and combinations, Pascal's triangle, inclusion-exclusion, and other counting methods.

The material is chosen so that the various themes of the camp weave together and introduce the campers to the history of our subject. The Pythagoras Level curriculum alternates yearly, so returning campers will enjoy seeing both familiar faces and new material.

## Euclid Level, Ages 9-11

All new campers age 9 to not yet 12 start in the Euclid Level, where they obtain a foundation for future learning. Campers who have previously participated in the Pythagoras Level also attend the Euclid level once they are nine years old. The curriculum of courses for the first year may cover geometry, number systems, and mathematical logic and methods of proof, such as the following past course descriptions:

**Euclidean Geometry**We work through the first book of Euclid's Elements, an ancient book of geometry that can be considered the book that founded mathematics. We study straightedge and compass construction, and we discover ourselves the proofs of many basic theorems. We discuss the gaps is Euclid's approach, and the modern foundations of Euclidean geometry by Hilbert.

**Voting Theory**We will explore the mathematics behind different voting methods and criteria for fair elections. We will explore the limitations of voting methods, including the proof of Arrow's Impossibility Theorem which states that no ranked voting method can satisfy all criterion desired for a "fair" election. We will explore non-ranked voting alternatives, such as instant run off voting.

**Number Theory**We introduce the division algorithm in order to study a variety of applications including different positional number systems, the divides relation, greatest common divisors and solutions to linear Diophatine equations using the extended Euclidean algorithm, modular arithmetic and divisibility rules. We will then delve deeper into modular arithmetic by investigating Fermat's little theorem, Wilson's theorem, and Euler's theorem as well as finding roots modulo n. We finish by discussing the Chinese remainder theorem. Along the way we will introduce various related puzzles and tricks.

**Methods of Proof**Starting with the three classical laws of thought (identity, non-contradiction, and excluded middle), the course progresses through propositional calculus and the deduction theorem to the method of direct proof and the method of indirect proof, the well-ordering principle, Peano's fifth axiom, the Fundamental Theorem of Mathematical Induction (FTMI), the equivalence of well-ordering and FTMI, and examples of proofs using the various methods.

## Gauss Level, Ages 10-11

After their Euclid year, campers who are invited to return are then placed in the Gauss Level. In this year, we offer two standard courses: algorithms, and mathematical functions and sets. Campers also attend a third course based on faculty expertise, such as symmetry, number theory, or combinatorics.

**Sets and Functions**Using naïve set theory, we will develop the concepts and properties of ordered pair, relation, and function. We will discuss Russell's paradox and the need for axioms. We will also learn about cardinals, ordinals, and the axiom of choice. We will likely conclude the course with the Cantor-Schroder-Bernstein Theorem.

**Algebraic Curves**We will study the geometry of zeroes of the polynomials in two variables over the reals. After studying linear polynomials we will learn how the quadratic polynomials can be classified. Useful methods like factorization or substitution of variables will be discussed and used to study higher degree curves. We will prove the Nullstellensatz for some simple cases. As an application we decide if Watt's curve includes a straight line segment or not, and how the situation can be improved by using 6-bar or 8-bar planar linkages.

**Enumerative Combinatorics**Enumerative has the same Latin root as the word number, and combinatorics is derived from the German and English words meaning combinations. This class will focus on the counting of combinations. Enumerative combinatorics questions are often in the form "How many different ways can you select a subset of objects from a larger group?" Characteristics of the desired subset and properties of the group of objects can vastly change the difficulty and complexity of the questions. We will explore counting in the contexts of probability and graph theory. We will to use the context of the board game Pandemic to motivate counting in a discrete mathematical modeling context.

**Visual Group Theory**We will focus on symmetry groups from the view of group actions by studying the symmetries of objects such as games (for instance Spinpossible and the Rubik's cube), rearrangements of cards, regular polygons, and light switches. After playing with Cayley diagrams and letting our intuition guide us, we will see the need for a more rigorous algebraic definition of symmetry groups, which will lead us to the study of groups more formally. We will then focus on properties of groups generalizing familiar arithmetic/algebraic laws and investigate Cayley tables. We will study cyclic groups, dihedral groups, direct products of $\Z_2$, and symmetric groups (as well as the hyperoctahedral groups), and finish with Cayley's theorem. Along the way, we will discuss one-to-one and onto functions, groups created by generators and relations, directed graphs, and the notion of subgroups and group isomorphism (these last two informally).

**Projective Geometry**We introduce homogeneous coordinates of points and lines of the Euclidean plane; then we add the ideal line to construct the real projective plane. We study a little linear algebra to use it as a tool. We prove the theorem of Desargues, and if time permits, the theorem of Pappus in the real projective plane. Then we change gears: we start the axiomatic study of projective planes. We discover the existence of finite geometries, and we prove many properties, including relations to Graeco-Latin squares.

## Conway Level, Age 11-12

If still age-eligible and again invited to return, campers are then placed in the Conway Level. In this year, we run a collection of courses created by the faculty related to advanced areas of mathematics, such as group theory, knot theory, topology, and finite geometry. The final topics depend on faculty expertise.

**Advanced Set Theory**In this course we will study the Zermelo-Frankel axiom system (together with the axiom of choice). We will carefully develop and explore theories for (a) unstructured sets and (b) well-ordered sets. In each setting, we will compare sizes and develop a version of arithmetic that includes addition and multiplication. If time allows, we'll recursively define the Aleph function and show that it has a fixed point.

**Fractals**We consider several geometric objects that are self-similar, including Koch's curve and snowflake, the Cantor set, the Sierpinski triangle and carpet, and the dragon curve. The notion of a Lindenmayer system will be defined. We will learn about computing the limit of a sequence and the sum of a series, in order to find the dimension of a fractal. We will introduce the set of complex numbers to study the Mandelbrot set and the Newton fractal as well.

**Symmetry and Orbifolds**We study and classify symmetries of the plane and sphere using topology.Along the way we will prove the classification of surfaces, study Euler characteristic, and enumerate orbifolds. As time permits we will consider the symmetries of the hyperbolic plane, geometrizing arbitrary surfaces, Cayley graphs of planar symmetry groups using "Van Kampen tilings", color

symmetries, covering spaces, and the fundamental group and 1st homology of surfaces.