Suppose we place checkers in the lower left corner of a Go board and wish to move them to the upper right corner in as few moves as possible, where the pieces move as in the game of Chinese checkers. Auslander, Benjamin, and Wilkerson in 1993 generalized this game for integer lattices and defined a measure of speed for a starting configuration of pieces. They proved that the maximum speed of any configuration is 1, and only three configurations, called"speed-of-light"configurations, attain this speed. We prove their conjecture that the maximum speed of a non-speed-of-light configuration is 2/3 in the 2-dimensional case, and present a framework that should extend to higher dimensions.

We present combinatorial proofs of identities inspired by the Hosoya Triangle.

We provide a combinatorial interpretation of the q-binomial and q-multinomial coefficients as counting weighted collections of tiled boards. Using this interpretation, we prove a new q-analogue to Lucas' Theorem and new q-analogues to identities on the sums of integer squares and cubes. Further proofs of known q-identities illustrate the use of proof elements including generating functions, recurrence relations, and sign-reversing involutions, all in the q-binomial context.

While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down permutations.

In a recent work, Baxter and Pudwell mentioned the following identity for the Fibonacci numbers F-n and noted that it can be proven via induction: For all n >= 1, F-2n = 1 . F2n-2 + 2 . F2n-4 + ... + (n - 1) . F-2 + n. We give a combinatorial proof of this identity and a companion identity. This leads to an infinite family of identities, which are also given combinatorial proofs.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsArthur BenjaminArthur Benjamin teaches at Harvey Mudd College and is a past editor of Math Horizons. He thanks Jay Cordes, Adam, Busis, and Bob Koca for valuable assistance.Joseph KisenwetherJoseph Kisenwether is a mathematics and game design consultant to the casino industry and founder of Craftsman Gaming. He describes his job as “the reason you can't win.”Ben WeissBen Weiss is the developer of the Frax app that allows users to navigate fractal images in real time. He has competed as a free diver for the USA in international competitions and has held his breath for over seven minutes. He works for Google in Southern California.

Summary We describe a “handy” method, due to John Conway, for quickly finding all relatively small prime factors of 3-digit and 4-digit numbers.

This chapter focuses on Hall's Theorem, introduced by British mathematician Philip Hall, and its connection to graph theory. It first considers problems that ask whether some collection of objects can be matched in some way to another collection of objects, with particular emphasis on how different types of schedulings are possible using a graph. It then examines one popular version of Hall's work, a statement known as the Marriage Theorem, the occurrence of matchings in bipartite graphs, Tutte's Theorem, Petersen's Theorem, and the Petersen graph. Peter Christian Julius Petersen introduced the Petersen graph to show that a cubic bridgeless graph need not be 1-factorable. The chapter concludes with an analysis of 1-factorable graphs, the 1-Factorization Conjecture, and 2-factorable graphs.

This chapter considers distance in graphs, first by providing an overview of some fundamental concepts in graph theory. In particular, it discusses connected graphs, cut-vertex and bridge, and bipartite graphs. It then addresses questions of the distance between locations in a graph and those locations that are far from or close to a given location. It also looks at dominating sets in graphs, focusing on the Five Queens Problem/Puzzle and the Lights Out Puzzle, before concluding with an analysis of the rather humorous concept of Erdős numbers, conceptualized by Hungarian mathematician Paul Erdős. According to this concept, for each mathematician A, the Erdős number of A is the distance from A to Erdős in the collaboration graph. Consequently, Erdős is the only mathematician with the Erdős number 0, whereas any mathematician who has coauthored a paper with Erdős has Erdős number 1.

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

This chapter considers a new type of graph coloring known as edge coloring. It begins with a discussion of an idea by Scottish physicist Peter Guthrie Tait that led to edge coloring. Tait proved that the regions of every 3-regular bridgeless planar graph could be colored with four or fewer colors if and only if the edges of such a graph could be colored with three colors so that every two adjacent edges are colored differently. Tait thought that he had found a new way to solve the Four Color Problem. The chapter also examines the chromatic index of a graph, Vizing's Theorem, applications of edge colorings, and a class of numbers in graph theory called Ramsey numbers. Finally, it describes the Road Coloring Theorem which deals with traffic systems consisting only of one-way streets in which the same number of roads leave each location.

This chapter considers a class of graphs called trees and their construction. Trees are connected graphs containing no cycles. When dealing with trees, a vertex of degree 1 is called a leaf rather than an end-vertex. The chapter first provides an overview of trees and their leaves, along with the relevant theorems, before discussing a tree-counting problem, introduced by British mathematician Arthur Cayley, involving saturated hydrocarbons. It shows that counting the number of saturated hydrocarbons is the same as counting the number of certain kinds of nonisomorphic trees. It then revisits another Cayley problem, one that involved counting labeled trees, and describes Cayley's Tree Formula and the corresponding proof known as the Prüfer code. It also explores decision trees and concludes by looking at the Minimum Spanning Tree Problem and its solution, Kruskal's Algorithm.

This chapter considers Eulerian graphs, a class of graphs named for the Swiss mathematician Leonhard Euler. It begins with a discussion of the the Königsberg Bridge Problem and its connection to Euler, who presented the first solution of the problem in a 1735 paper. Euler showed that it was impossible to stroll through the city of Königsberg, the capital of German East Prussia, and cross each bridge exactly once. He also mentioned in his paper a problem whose solution uses the geometry of position to which Gottfried Leibniz had referred. The chapter concludes with another problem, the Chinese Postman Problem, which deals with minimizing the length of a round-trip that a letter carrier might take.

This chapter considers problems of whether a graph can be decomposed into certain other kinds of graphs, primarily cycles. It begins with a background on nineteenth-century mathematician Thomas Penyngton Kirkman and the problem he invented known as Kirkman's Schoolgirl Problem, stated as: How many triples can be formed with x symbols in such a way that no pair of symbols occurs more than once in the triple? This is followed by a discussion of the Steiner triple system, the relationship between cyclic decomposition problems and a problem called Alspach's Conjecture, graceful graphs, and the Graceful Tree Conjecture. The chapter concludes with an analysis of the puzzle dubbed Instant Insanity and how graphs can be utilized to solve it.

This chapter focuses on oriented graphs and their use to represent a sports tournament where assigning a direction to an edge represents the defeat of one team by another. It first considers the 1941 book What Is Mathematics?, co-authored by Herbert Ellis Robbins and Richard Courant, before discussing Robbins's Theorem in terms of graph theory. According to Robbins's theorem, it is possible to repair any one street and still be able to travel between any two points in the city discussed by Robbins if and only if it is possible to convert all streets of the city to one-way streets and travel (legally) between any two points. The chapter also examines the best known class of oriented graphs, the orientations of complete graphs, and their application to round robin tournaments. Finally, it describes the King Chicken Theorem and how various voting techniques can result in often surprising outcomes.

This chapter provides an introduction to graphs, a mathematical structure for visualizing, analyzing, and generalizing a situation or problem. It first consider four problems that have a distinct mathematical flavor: the Problem of the Five Princes, the Three Houses and Three Utilities Problem, the Three Friends or Three Strangers Problem, and the Job-Hunters Problem. This is followed by discussion of four problems that are not only important in the history of graph theory, but which led to new areas within graph theory: the Königsberg Bridge Problem, the Four Color Problem, the Polyhedron Problem, and the Around the World Problem. The chapter also explores puzzles and problems involving chess that have connections to graph theory before concluding with an overview of the First Theorem of Graph Theory, which is concerned with what happens when the degrees of all vertices of a graph are added.