Drawing a Triangle on the Thurston Model of Hyperbolic Space
by Curtis D. Bennett, Blake Mellor, and Patrick D. Shanahan
In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.
A Survey of Euler’s Constant
by Thomas P. Dence and Joseph B. Dence
The mathematical constant, denoted by γ, and referred to as Euler’s constant, with value approximately 0.5772156, is not nearly as well known as the constants π and e, but is still important enough to warrant serious discussion. We present here a variety of instances in mathematics where γ occurs, and also a number of different expressions (typically involving either an integral, a series, or a limit) that all represent this fascinating number. An extensive reference serves to provide interested readers with alternate sources.
The Multiplication Game
By Kent E. Morrison
The Multiplication Game is a two-person game in which each player chooses a positive integer without knowledge of the other player’s number. The two numbers are then multiplied together and the first digit of the product determines the winner. Rather than analyzing this game directly, we consider a closely related game in which the players choose positive real numbers between 1 and 10, multiply them together, and move the decimal point, if necessary, so that the result is between 1 and 10. The mixed strategies are probability distributions on this interval, and it is shown that for both players it is optimal to choose their numbers from the Benford distribution. Furthermore, this strategy is optimal for any winning set, and the probability of winning is the Benford measure of the player’s winning set. Using these results we prove that the original game in which the players choose integers has a well-defined value and that strategies exist that are arbitrarily close to optimal. Finally, we consider generalizations of the game in which players choose elements from a compact topological group and show that choosing them according to Haar measure is an optimal strategy.
Mini-Sudokus and Groups
By Carlos Arcos, Gary Brookfield, and Mike Krebs
Using a little group theory and ideas about equivalence relations, this article shows that there are only two essentially different 4 by 4 Sudoku grids. This can be compared with the 5,472,730,538 essentially different 9 by 9 Sudoku grids found by Jarvis and Russell with the aid of a computer algebra system.
Gershgorin Disk Fragments
by Aaron Melman
Eigenvalue inclusion regions for a general complex matrix can be found by forming the Gershgorin disks, centered at the diagonal elements of the matrix. We give a standard proof of Gershgorin’s theorem and show how it can be continued in a natural way to derive a lesser known result obtained by Parodi and Schneider, which uses fragments of the Gershgorin disks. We provide some examples, including an application to the location of polynomial zeros.
Cosets and Cayley-Sudoku Tables
by Jennifer Carmichael, Keith Schloeman, and Michael B. Ward
The popular Sudoku puzzles are 9 by 9 tables divided into nine 3 by 3 sub-tables or blocks. Digits 1 through 9 appear in some of the entries. Other entries are blank. The goal is to fill the blank entries with digits 1 through 9 in such a way that each digit appears exactly once in each row and in each column and in each block. Cayley tables are group operation tables. As such, each group element always appears exactly once in each row and in each column, two thirds of being a Sudoku-like table. Cayley-Sudoku tables are Cayley tables arranged in such a way as to satisfy the additional requirement. Namely, the Cayley table may be divided into blocks with each group element appearing exactly once in each block. We show three ways to construct nontrivial Cayley-Sudoku tables for finite groups, present a Cayley-Sudoku puzzle to solve, and give several suggestions for additional investigations.
Proof Without Words: Mengoli's Series
by Ángel Plaza
by Dimitrios Kodokostas
A triangle equalizer is a line bisecting both its area and perimeter. We provide a detailed account of equalizer locations, showing that there exist triangles with exactly one, two, or three equalizers, but no more. Triangles with exactly two equalizers are quite rare: Their smallest angle is less or equal to a particular angle of approximately 49 degrees, and the angle next in size is the unique root of a trigonometric equation in a specific interval depending on the smallest angle.