Sunday, January 22, 2017

Abstracts of the papers in “Journal of Recreational Mathematics” Volume 10, Number 2, 1977-78

Abstracts of the papers in “Journal of Recreational Mathematics” Volume 10,  Number 2, 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.

“A ‘Practical’ Approach to Egyptian Fractions,” by Paul J. Campbell
 Egyptian fractions are ratios where the numerator is one and if they are the only fundamental numbers you have, the challenge becomes representing all other fractions using only Egyptian fractions. Since it is possible to have more than one representation of a fraction using Egyptian fractions, the issue becomes the creation of a “minimal” representation.
 There are two ways in which this can be defined, the least number of fractions or the smallest largest denominator. In this paper some algorithms for determining the minimal representation are given. 

“Complete Threading of Surface Models,” by Michael Gregory
 One way in which complex three-dimensional figures can be modeled is by constructing a rigid skeleton and then running strings from location to location. In this paper, algorithms used to design the threading are examined and analyzed. 

“Related Magic Squares With Prime Elements,” by Gakuho Abe
 It is possible to form a sequence of consecutive primes of any length and a sequence of the proper length can be placed in an n × n grid. The purpose of this paper is to present some magic squares and magic cubes formed from sequences of consecutive primes. 

“Polyomino and Polyiamond Problems: Part II,” by Wade E. Philpott
 A hexiamond is formed by joining the edges of six identical equilateral triangles and there are twelve of them. Heptiamonds are formed by joining the edges of seven identical equilateral triangles. In this paper, a series of figures constructed using the hexiamonds and heptiamonds are given.

“Analysis of a Pattern,” by Samuel Yates
 The infinite set of integers having the form n nines, n – 1 zeros and ending with a single one are defined as the “Beiler numbers.” The first three in the sequence are 91, 9901 and 999001. For the first few Beiler numbers they alternate between being composite and prime and then that pattern ends.
 In this paper an explanation of why the pattern is broken and the majority of the Beiler numbers are then composite is given. It is due to the larger numbers having repunit factors, which are numbers constructed of all ones. 

“Digital Fractions: II,” by Stewart Metchette
 For the purposes of this paper, a digital fraction is one of the form 1 / n that is equal to the 9-digit fraction ABCD / EFGHI or the corresponding fraction made with ten digits,
ABCDE / FGHIJ. No digit is repeated in the fractions, although the value of n can. A program was written to search for solutions for various values of n and this paper is a report of the results.

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