Monday, January 23, 2017

Abstracts of the papers in “Journal of Recreational Mathematics” Volume 1, Number 1, 1968

Abstracts of the papers in “Journal of Recreational Mathematics” Volume 1, Number 1, 1968

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

“Magic Designs,” by Robert B. Ely, III

 The definition of a magic design is as follows:

A design with N parts is said to be magic, if those parts can be labeled with the numbers 1 to N so that the labels of each of a number of identical sub-designs give the same constant total.

 A magic square, which is an m x m grid where each row and column has the same sum  is the most widely known magic design.
 In this paper, additional magic designs, such as a triangle, cube, a grid made of hexagons, a triangular grid and the faces of a dodecahedron are analyzed.

“Counting Planar Maps,” by W. T. Tutte

 The paper opens with the examination of a standard die as an object having six faces. Such an object is called a hexahedra. There are seven convex objects having exactly six faces and three that are concave. Schlegel diagrams are used to convert the polyhedra into planar graphs for further analysis. Planar diagrams that can be defined by 3-connected graphs are called c-nets.
A table of the number of c-nets for each number of edges for n from 4 through 25 is given along with the explicit formula used to compute the numbers.

“Infinite Geometry,” by Donald L. Vanderpool

 While Euclidean geometry is performed on a plane infinite in two directions, triangles are described as being finite. In this paper, the lines that form the triangles are geodesic’s in Einstein’s 4-dimensional space. This leads to three-sided figures where the sums of the angles are vastly different from the standard 180 degrees.

“A Recurrent Operation Leading to a Number Trick,” by Charles W. Trigg

 For each two-digit number S, the digits are written in reverse order. This is added to the sum of the digits of S and then the sum is reduced modulo 100. The computations are then repeated.
 Under this operation, the fifty-two odd two-digit numbers form a series in which the last 6 numbers form a closed loop.

 “Alphametics,” by J. A. H. Hunter

 This paper re-introduces the letter-arithmetic puzzle of alphametic and gives a brief description of the logic one uses to solve them. The example problem is a division alphametic.

“The Witch of Agnesi,” by Harold D. Larsen

 The Witch of Agnesi is a curve defined by the equations:

             y( a2 + x2) = a3 or x = a(cot θ), y = a(sin θ)2.

 The history and some of the major properties of this curve are examined.

“Curiosa for 1968,” by Charles W. Trigg

 This paper gives several ways that 1968 can be written using conventional math symbols and the nine decimal digits.

“Pentomino Farms,” by Fr. Victor Feser, OSB

 The 12 pentominoes are formed from five squares being connected edge-to-edge. A pentomino farm is a rectangular shape made by the set of pentominoes that has enclosed open space(s). Several of the most basic are presented.

“Squares with 9 and 10 Distinct Digits,” by T. Charles Jones

 In this paper, two computer generated tables are given. The first contains all squares with nine distinct digits and the second all squares with ten distinct digits.

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