Monday, March 21, 2016

Abstracts to the papers that appeared in "Journal of Recreational Mathematics 37(1)"

Abstracts to the papers that appeared in Journal of Recreational Mathematics 37(1)

Simple – But Little Known – Methods of Generating Pythagorean Triples

Owen O’Shea

 In this paper, the special form of Pythagorean triples (a, b and c where a*a + b * b = c*c) and
the hypotenuse is one greater than  the longest leg is examined. A pattern emerges for the
generating of such triples that involves the use of triangular numbers.

abc – Triples and Their Statistical Distribution

Myriam Le Boulicaut and Bernard Haussy

 This article describes two new methods for finding abc-triples as well as statistical considerations on the distribution of the different values of α found by the algorithms which were programmed. α represents the quality of a triple; the higher the quality, the better the triple. A beginning search for abc-triples with low values of α will also be introduced.  

On a Conjecture of Dey and De
Robert J. MacDawson
Saint Mary’s University

In Volume 36 of Journal of Recreational Mathematics, Dey and De conjecture that any two squares of primes, each ending (when expressed in decimal notation) in 1, differ by a multiple of 120; and that any two squares of primes, each ending in 9, differ by a multiple of 40. In this note, we prove this conjecture, replacing primality with a weaker condition.  A related result for fourth powers is also given.

When is the Honeymoon Over for Baseball’s New Stadiums?
Mark B. Whelan and Paul M. Sommers
Middlebury College

For the twelve newly-built Major League Baseball stadiums opened between 2000 and 2009, the authors compare average home attendance during the last year in the old stadium to the average home attendance the first, second, and third years in the new stadium.  For ten of the teams, the average home attendance in the opening year was significantly higher than the year before.  Nine  of these ten  teams did not enjoy a significantly higher attendance increase in their second year.  And  in only three of the nine cases was the average attendance higher in the third year than the year before the new stadium opened.

Rezaei Method for Construction of Magic Squares of All Even Orders
Saeed Rezaei Toroghi

An elementary magic square is a square matrix in which every row, column and diagonal sums to a magic constant and in which the elements are consecutive integers starting from 1. In order to construct such squares, different methods have been devised for every type of magic square based on their number of rows or columns. The problems was that there has not been a universal algorithm to solve all magic squares with an even number of rows or columns. Previous methods have provided different methods for magic squares with a doubly even and singly even number of rows or columns. Here, a universal algorithm is provided to construct all magic squares with an even number of rows and columns. 

A Squirtgun Battle
Stephen Portnoy
University of Illinois at Urbana-Champaign

  In his book, Mathematical Puzzles: a Connoisseur's Collection, Peter Winkler attributes the following problem to the Sixth All Soviet Union Mathematics Competition in Voronezh, 1966: `"An odd number of soldiers are stationed in a field, in such a way that all pairwise distances are distinct. Each soldier is told to keep an eye on the nearest other soldier. Prove that at least one soldier is not being watched."  In  Allan Gottlieb's ``Puzzle Corner'' (in Technology Review,  2008), Jerry Grossman  changes `"soldiers" to "children with squirtguns" (again at distinct distances and shooting at the nearest other child) and adds the problem to show that if n is even  then there are configurations of children where all get wet. In the course of solving these problems, the following generalization seemed intriguing and appears to be new: Given n children, what is  the greatest number of children than can remain dry? A special case of the problem is to consider the two nearest children (A and B) and to ask for the largest number of other children that can be added so than only A and B get wet. For the special case, it turns out that a maximum of 7 dry children can be added. It is not too difficult to use this result to solve the  general problem for n children.

Amidakuji and Games
Steven T. Dougherty and Jennifer Franko Vasquez

We describe two games based on Japanese ladders.  The games are related to finding the minimum number of a certain type of transpositions to describe a given permutation.  The mathematics of both games is explored. 

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