Judith D. Sally, Northwestern University, Evanston, IL, and Paul J. Sally, Jr., University of Chicago, IL

The Four Numbers Problem

Introduction

The Four Numbers Game Rule

Symmetry and the Four Numbers Game

Does Every Four Numbers Game Have Finite Length?

Games With Length Independent of the Size of the Numbers

Long Games

Some Formal Notation

Constructing Long Games

The Tribonacci Games

Computation of L(Tn)

Upper Bounds for Lengths of Games

The Length of the Four Real Numbers Game

Linear Algebra Comes Into Play

Construction of a Four Numbers Game of Infinite Length

Construction of All Four Numbers Games of Infinite Length

The Probability that a Four Numbers Game Ends in n Steps

The k-Numbers Game

Bibliography

Introduction

Right Triangles

Pythagorean Triples

Sums of Squares

The Two Squares Theorem

Characterization of the Length of the Hypotenuse of an Integer Right Triangle

The Number of Representations of n as a Sum of Two Squares

Rational Right Triangles

Congruent Numbers

Equivalent Definitions of Congruent Number

1, 2, and 3 Are Not Congruent Numbers

Rational Right Triangles and Certain Cubic Curves

Elliptic Curves

The Abelian Group of Rational Points on an Elliptic Curve

En(Q) and Congruent Numbers

Bibliography

Introduction

Geometric Shapes as Lattice Polygons

Properties of Lattice Polygons in the Plane

Embedding Regular Polygons in a Lattice

Regular Lattice n-gons

Which Positive Integers Are Areas of Lattice Squares?

Basic Algebraic and Geometric Tools

Dissection of a Lattice Polygon Into Lattice Triangles

The Algebraic Structure of the Lattice Z2

The Isometry Group of a Lattice

Pick’s Theorem

First Proof

From Euler to Pick

Visible Lattice Points

Pick’s Theorem for nP

Applications of Pick’s Theorem

Lattice Triangles T with I(T) = 0 and 1

Farey Sequences

Lattice Points In and On a Circle

Integer Points in Bounded Convex Regions in R2

Convex Plane Regions and Integer Points

An Application of Pick’s Theorem to Bounded Convex Regions

Minkowski’s Theorem in R2

Embedding Regular Plane Polygons as Lattice Polygons in Rk

Lattice Hypercubes

Minkowski’s Theorem in Rk

Ehrhart’s Theorem

Convex Polytopes

Ehrhart’s Theorem for a k-Simplex

The Coefficients of the Ehrhart Polynomial

Bibliography

Introduction

Introduction to Approximation Theory

Properties of Rational Numbers Close to a Real Number

An Interesting Example, Part I

Dirichlet’s Theorem

An Interesting Example, Part II

Hurwitz’s Theorem

Liouville’s Theorem

Statement and Proofs of Liouville’s Theorem

Liouville’s Theorem and Transcendental Numbers

The Thue-Siegel-Roth Theorem

Introduction

Thue’s Theorem

Roth’s Theorem

The Approximation Exponent

An Interesting Example, Part III

An Application to Diophantine Equations

What About Transcendental Numbers?

Bibliography

Introduction

Dissection and Area

Basic Properties of Dissection

Polygons of Equal Area

Dissection in Three Dimensions

The Angles of a Polyhedron

The Dehn Invariant

A Solution of Hilbert’s Third Problem

Congruence by Finite Decomposition and Equidecomposability

Hausdorff’s Paradox

The Banach-Tarski Paradox

Equidissectability and Equidecomposability

Squaring the Circle

Borsuk’s Problem

Borsuk’s Conjecture in the Plane

Borsuk’s Conjecture in R3

Closed Convex Sets with Smooth Boundary

Bibliography