List of abstracts of the papers that appeared in "Topics in Recreational Mathematics Volume 6" edited by Charles Ashbacher

List of abstracts of the papers that appeared in Topics in Recreational Mathematics Volume 6 edited by Charles Ashbacher, ISBN 978-1530004225

The Impact of Free Agency on NHL Player Performance

Emily A. Fluke, Paul M. Sommers

Department of Economics
Middlebury College
Middlebury, Vermont 05753
psommers@middlebury.edu

Abstract

The authors use “point shares” to assess the impact of free agency on National Hockey League player performance by dividing restricted and unrestricted free agents in 2013-14 into two groups, players who stayed with their same team and those who switched to another team.  Restricted free agents (and to a lesser extent, unrestricted free agents) who play for a new team perform worse and those who play for their previous team perform marginally better.  The study underscores the importance of (i) separating restricted from unrestricted free agents and (ii) noting that staying with the same team or playing for a new one differentially affects free agent performance.  

Some Conjectures on the Carmichael Numbers

Marius Coman, Charles Ashbacher

Abstract

 In his book “Two Hundred Conjectures and One Hundred and Fifty Open Problems On Fermat Pseudoprimes” Marius Coman states several conjectures about the Carmichael numbers. The purpose of this short paper is to state some of them that seem amenable to computer analysis.

Linear Algebra Properties of Magic Squares

Hossein Behforoot

Mathematics Department

Utica College

Utica, New York 13502

hbehforooz@utica.edu

www.utica.edu/hbehforooz

Abstract

 Overall, every magic square is a very special square matrix and in this article we are going to show some interesting linear algebra properties of these magic square matrices. There are some published short articles on this subject but they are not complete papers with all properties in one article.

The Importance of Winning Draw Controls in Women’s Lacrosse

Alexandra L. DeMarco, Zoe M. Loveman, Paul M. Sommers

Department of Economics
Middlebury College
Middlebury, Vermont 05753
psommers@middlebury.edu

Abstract

 The authors examine the individual box scores of all men’s and women’s lacrosse games in 2013, 2014 and 2015 for the eleven schools in the New England Small College Athletic Conference to assess the importance of winning face offs to winning games. In men’s lacrosse, there is no relationship between face offs won and games won.  In women’s lacrosse, however, there is a strong direct relationship between draw controls (as face offs are called in women’s lacrosse) and games won.  The authors then use regression analysis to find (for each of the eleven schools) the minimum percentage of draw controls won needed to win a game.

Radical Axis of Lemoine Circles

Ion Patrascu

Professor “Frații Buzești” National College,

Craiova, Romania

Florentin Smarandache

Professor, New Mexico University, USA

Abstract

 In this short paper, a theorem stating that the radical axis of of the Lemoine circles of a triangle is perpendicular to a line raised on the symmedian is proven.

Word Hypercubes are Fun, NP-Hard, and In General Undecidable

Barry Fagin

Leemon Baird

Abstract

 Word-hypercubes are a generalization of word squares for more than 2 dimensions.  Given a dictionary of words of length n, and a d-dimensional hypercube partitioned into n^d spaces of dimension d, can letters be placed into all spaces so that words from the dictionary are formed when reading unidirectionally?  We show that this problem is NP-complete, and also give examples of both new word squares and the word hypercube of highest dimensionality known to the authors.

Mr. Browne and the Dance of Yu:Constructing a Normal Magic Square of Order 3ⁿ

Frank J. Swetz

Abstract

 The normal magic square of order three has fascinated viewers and perplexed problem solvers for centuries. Its origins can be traced to ancient China where it was known as the Luoshu [Luo river writing or document] and used as a subject for ritual and numerological manipulation. Over time, the Chinese developed a series of magic squares but these remained ceremonial devices devoid of mathematical theory. In 1917, magic square enthusiast C. A. Browne published a twenty-seven order normal magic square embedded with mystical properties. A mathematical analysis of Mr. Browne’s square reveals a relationship to the Luoshu. Through the use of a judicious system of partitioning and repeated iteration of the Luoshu pattern, it appears that for any N, a positive, integer, a normal magic square of order 3ⁿ can be constructed.

Geometry and Design of Equiangular Spirals

Kostantinos Myrianthis

myrian@ath.forthnet.gr

Abstract

 In an equiangular spiral, "the whorls continually increase in breadth and do so in a steady and unchanging ratio... It follows that the sectors cut out by successive radii, at equal vectorial angles, are similar to one another in every respect and that the figure may be conceived as growing continuously without ever changing its shape the while" as stated by Sir D'Arcy W. Thompson and quoted in [1, p.125]. I was fascinated since my early years with the shape of spirals and all their versions in nature. The mathematical modeling of them became a very attractive topic of study and research for me and more specifically, the geometrical conditions under which any quadrangle or triangle can be fitted into similar copies of itself and form an equiangular spiral. This formation gives the impression of a digital form of spiral, where every digit is a triangle or quadrangle following similarity laws, which can allow a multiplicity of design capabilities. The essence of this work appears in the present article and is related with the geometry and the design characteristics of equiangular spirals.