Sunday, January 22, 2017

Abstracts of the papers in “Journal of Recreational Mathematics” Volume 10, Number 3, 1977-78

Abstracts of the papers in “Journal of Recreational Mathematics” Volume 10,  Number 3, 1977-78

 Given the elapsed time since these papers appeared and that there were no abstracts with the originals, nearly all of the items in this list were written by Charles Ashbacher.

“Rational Transposables,” by Steven Kahan
 This abstract is a paraphrase of the first paragraph of the paper.
 A complete characterization of k-transposable and k-reverse-transposable integers have been given in previous publications for all possible integral values of k, 2 ≤ k ≤ 9. We now reconsider the definitions of these terms, relaxing the restriction that k be integral.

“Antimagic Pentagrams With Consecutive Line Sums,” by Charles W. Trigg
 A pentagram is a regular five-pointed star produced from line segments of equal length. There are five points and five intersections of the line segments that form a regular pentagon. Numbers can be placed on each point and intersection, placing four numbers on each segment. If the sum of the numbers on each segment are the same then the pentagram is magic, if all the sums are different it is said to be antimagic and there is also the potential for the lines sums to be consecutive integers. Searching for such pentagrams is the topic of this paper. 

“The ‘Betsy Ross’ Star,” by Robert T. Kurosaka
 An interesting historical legend is that George Washington wanted to have five-pointed stars on the American flag, but considered the construction of such stars too complex. Supposedly, seamstress Betsy Ross quickly disproved that supposition by folding a square of paper, making a single cut and then demonstrated the star. This paper describe how she may have done that. 

“Packing Boxes With Congruent Polycubes,” by Andrew L. Clarke
 A polycube is a solid formed by putting unit cubes together by their faces. Using them to create figures, specifically rectangular parallelpipeds (boxes) has been an active area of recreational mathematics. If an odd number of identical pieces can be used to form a box then the polycube is said to be odd and if an even number can be used to construct a box then it is said to be even. Examples of such constructions and the techniques used to achieve them are the purpose of this article. 

“Amicable Pairs of Euler’s First Form,” by Patrick J. Costello
 A pair of integers is said to be amicable if each is the sum of the proper divisors of the other. As it was with so many things, using number theory techniques, Leonhard Euler discovered many such pairs.
 Euler’s first form is A = Epq and M = Er, where p, q and r are distinct primes not dividing the common factor E. Using this form, the author was able to discover additional amicable pairs as well as determine conditions when numbers cannot be amicable. 

“Binary Tic-Tac-Toe,” by John Michael Schram
 Binary tic-tac-toe is played on the standard 3 × 3 grid where the rows and columns are each assigned the values of the first three powers of two. Several different two-player games can then the played, based on changing the desired goal for victory using the powers of two. Some games of this form are analyzed. 

“Number Patterns in More Than One Dimension, Part I,” by Doug Engel
 The traditional way of writing numbers is to write the digits left to right, which means that each sequence of digits can represent only one number. However, the digits can also be written in a square array, which allows for rotations and reflections. This leads to the creation of new sets of patterns, which are analyzed when the numbers are in binary.

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