# 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.

## 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:

**Geometry from Euclid's***Elements*- We work through book 1 of Euclid's
*Elements*the ancient classic of mathematics. This course connects geometry to the physical through the use of ruler and compass. By following the structure of axioms and propositions given in*Elements*, we see how a complex world of behavior can be understood and explored by painstaking construction from a very simple collection of rules (the axioms). This idea is now the heart of modern mathematics.

- We work through book 1 of Euclid's
**Number Systems**- We consider representations of integers, fractions, and decimals in different exotic bases (like 3/2, -4, and Fibonacci), and the resulting arithmetic, as well as divisibility properties, divisors, and modular arithmetic.

**Mathematical Logic and 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.

**Algorithms**- This course is an introduction to the mathematical ideas at the heart of computer science. In an offline environment, we make ourselves into computers following steps to solve problems, and working out how to prove that our problem-solving techniques are correct and efficient.

**Mathematical Functions and Sets**- Using naïve set theory, this course describes the concepts and properties of ordered pair, relation, and function. Also covered: numbers, cardinals, and ordinals; Russell's paradox; the axiom of choice and axiomatic set theory. It concludes with the Cantor-Schroder-Bernstein Theorem.

## Conway Level, Age 11

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.