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Mathematics Magazine - February 2018

Mathematics Magazine February 2018 Cover

In this first issue of the new year, Benesh defines a metric on S4 based on the music video for “Gimme Sympathy” by the band Metric, while Befumo and Lenchner extend a classic Solomon Golomb theorem into higher dimensions.

Two articles in this issue have a historical theme: Thomas presents an understanding of the First Book of Spherics, while Nordgren offers a method for constructing Franklin squares. In other articles, Ahmed and Salleby derive the volumes of hyper-ellipsoids, and Mynard uses compactifications to topologically distinguish the punctured plane from the plane. Overgaard proves the optimality of an equation on minimizing the elongation of a heavy rope. Bessenyei and Szabó give a general solution to a functional equation from a competition textbook.

Caicedo and Shelton introduce a new puzzle from Lai Van Duc Thing: Partiti. Proofs without words, problems and solutions, and reviews complete the issue.

Michael A. Jones, Editor

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Vol. 91, No. 1, pp. 1 – 80

Articles

Letter from the Editor

p. 2.

Michael A. Jones

DOI: 10.1080/0025570X.2018.1404825

An Appreciation of the First Book of Spherics

p. 3.

R. S. D. Thomas

This paper offers an understanding of the contents of the first book of spherics, whoever wrote it—a rational reconstruction of what it may once have said that goes beyond the historical evidence, which is the second-century BCE Spherics of Theodosios. The reconstruction is done by bringing to the fore what is glossed over and adding the missing conclusion.

DOI: 10.1080/0025570X.2017.1404798

Distinguishing the Plane from the Punctured Plane Without Homotopy

p. 16.

Frédéric Mynard

After an informal short introduction to topology, I consider the problem of distinguishing the punctured plane from the plane topologically. I propose an alternative argument to the classical use of homotopy, relying instead on compactifications. Starting from the classical example of the stereographic projection realizing the sphere as the one-point compactification (a term that I explain) of the plane, I observe that the pinched sphere is the one-point compactification of the punctured plane. As the sphere is disconnected by a simple closed curve while the pinched sphere does not need to be, this provides an intuitive argument to distinguish topologically the plane from the punctured plane, without (explicitly) using homotopy.

DOI: 10.1080/0025570X.2017.1404797

Of Puzzles and Partitions: Introducing Partiti

p. 20.

Andrés Eduardo Caicedo and Brittany Shelton

We introduce Partiti, the puzzle that will run in this Magazine this year, and use the opportunity to recall some basic properties of integer partitions.

DOI: 10.1080/0025570X.2018.1403233

Partiti Puzzle

p. 23.

Lai Van Duc Thinh

How Franklin (May Have) Made His Squares

p. 24.

Ronald P. Nordgren

Franklin squares of order 8k are constructed by Euler's composite method with specified forms for the two orthogonal auxiliary squares. Two types of formulas are given for elements of the auxiliary squares that are shown to be orthogonal and to satisfy Franklin's three sum conditions. Squares of order 8 and 16 agree with Franklin's published squares and those of order 24 and 32 agree with squares previously constructed by a direct method. It is shown that Franklin squares can be transformed to pandiagonal magic squares in two ways but the converse is not true in general.

DOI: 10.1080/0025570X.2017.1404789

The Metric Metric on S4

p. 33.

Bret J. Benesh

We define a metric on the symmetric group on four elements based on the music video for “Gimme Sympathy” by the band Metric. In the video, the four band members rearrange themselves according to a set of permutations. We extend this set to include inverses and use this to define a metric on the group. This gives us the aptly named Metric metric on S4.

DOI: 10.1080/0025570X.2018.1405697

A Functional Equation View of an Addition Rule

p. 37.

Mihály Bessenyei & Gréta Szabó

Motivated by a problem posed in a competition textbook, we give the general solution of a functional equation related to the addition theorem of the hyperbolic tangent function.

DOI: 10.1080/0025570X.2018.1403230

Proof Without Words: On Sums of Squares and Triangles

p. 42.

Andrzej Piotrowski

We visually display a relationship between sums of squares and the sum of an even number of triangular numbers. Connections to some proofs without words appearing in the literature are briefly discussed.

DOI: 10.1080/0025570X.2018.1404885

On Volumes of Hyper-Ellipsoids

p. 43.

Shahnawaz Ahmed & Elias G. Saleeby

Volumes of objects in higher dimensions are of interest in higher-dimensional geometry. In this note, we give a sampling of some of the topics that are considered in this field. We first provide an alternative proof for a formula for the volumes of generalized super-ellipsoids that was obtained by Dirichlet in the 19th century. Then we employ a geometric Monte Carlo method that allows us to estimate hyper-volumes numerically. We then end the presentation with a brief discussion on the volumes of revolution in higher dimensions.

DOI: 10.1080/0025570X.2018.1404834

Proof Without Words: Three Arctangent Identities

p. 51.

Ángel Plaza

Visual proof of three arctangent identities involving arctan(√2 - 1) and arctan(√2 + 1).

DOI: 10.1080/0025570X.2018.1404888

Designing for Minimum Elongation

p. 52.

Niels Chr. Overgaard

We reconsider the variational problem of finding the shape of a vertically hanging rope such that its elongation, due to the rope's own weight and that of a load attached at its lower end, is minimum. The known solution is recalled and the missing proof of optimality is supplied.

DOI: 10.1080/0025570X.2017.1404790

Tiling One-Deficient Rectangular Solids with Trominoes in Three and Higher Dimensions

p. 62.

Arthur Befumo and Jonathan Lenchner

A classic theorem of Solomon Golomb’s states that if you remove a square from a chess board of size 2N × 2N then the resulting board can always be tiled by L-shaped trominoes (polyominoes of three squares). We show that if you remove a cube (hyper-cube) from a board of size K1 × ⋅⋅⋅ × KN, where K1 ⋅⋅⋅ KN ≡ 1(mod 3) , for N ⩾ 3, and at least three of the Ki > 1, then the remaining board can always be tiled by solid L-shaped trominoes. This extends 2D results of Chu and Johnsonbaugh from the 80s and results of Starr’s on 3D cubical boards from 2008. We also study the analogous problem for straight trominoes, showing that the same types of boards are never generically tilable (i.e., tilable regardless of square/cube/hypercube removed) using straight trominoes.

DOI: 10.1080/0025570X.2017.1404796

Proof Without Words: Products of Odd Squares and Triangular Numbers

p. 70.

Brian Hopkins

We visually demonstrate an identity equating the product of an odd number squared and a triangular number to a difference of triangular numbers.

DOI: 10.1080/0025570X.2017.1409534

Problems and Solutions

p. 71.

Proposals, 2036-2040

Quickies, 1077-1078

Solutions, 2011-2015

Answers, A1077-A1078

DOI: 10.1080/0025570X.2018.1411650

Reviews

p. 78.

Mathematics for logical thinking? Mathematics in reverse

DOI: 10.1080/0025570X.2017.1411696

Dummy View - NOT TO BE DELETED