Saturday, January 21, 2017

Abstracts of the papers that appeared in “Journal of Recreational Mathematics, Volume 1, Number 3, 1968"

Abstracts of the papers that appeared in “Journal of Recreational Mathematics, Volume 1, Number 3, 1968"

Abstracts were not included with the original papers, so all of the abstracts were written by Charles Ashbacher.

 “Of Knights and Cooks, and the Game of Checkers,” by Solomon W. Golomb
 A knight’s tour on a standard chessboard is where a knight is placed on a square and then moved so that it visits every square with no repeats. If the knight ends up at the starting square it is known as a closed tour and if it does not it is an open tour.
 In this paper a variety of related problems are considered. Knight’s tours on rectangles of various sizes and ways in which a set of knights can be placed on a board with no knight attacking any other are examined.
 A hybrid game of chess and checkers called “cheskers” that is played only on the black squares is examined. This game was invented by Golomb and a new piece called a “cook” had to be created as a knight cannot travel on this board. The cook moves three squares in one direction and one in the other so that it always remains on a black square. Cook’s tours and the placing of non-attacking cooks problems are also considered. 

“A Magic Square,” by William J. Mannke
 This paper presents an eight-by-eight magic square constructed by the numbers 1 through 64 where consecutive numbers appear in adjacent squares. In this case adjacent means across, down and diagonally. 

“Uncrossed Knight’s Tours,” by L. D. Yarbrough
 Knight’s tours on boards of various sizes from three-by-three to eight-by-eight where the paths do not cross each other are given.

“Doodling With Numbers,” by J. A. H. Hunter
 This paper contains a set of identities where sums and differences yield the value one and a set of sums where two identical representations in different bases yield a value where the same representation is in a different base.

“Generalized Fibonacci Numbers and the Polygonal Numbers,” by V. E. Hoggatt, Jr.
 The generalized Fibonacci numbers are defined by the recurrence relation
S(1) = 1, S(2) = 1, S(3) = 2, . . . , S(r) = 2^(r-2), S(r+1) = 2^(r-1).
Some power and summation identities for the generalized Fibonacci numbers are given.
The polygonal numbers are defined by the recurrence formula
P(1,r) = 1, P(n + 1,r) = P(n,r) + n(r – 2) + 1,
where r >= 3 is the number of sides of the polygon. Additional properties of these numbers are then described. 

“Nine-Digit Determinants Equal to 3,” by Charles W. Trigg
 The nine positive digits can be placed into a three-by-three array in 9! ways. It is then possible to compute the determinants of these arrays and the emphasis here is on the arrays where the determinants are equal to three. 

“From Forests to Matches,” by Ronald C. Read
 Graph theory is the area of mathematics where figures are made by using a set of nodes and a set of connections between them. The connections can be drawn as either straight or curved segments, depending on the contextual need. The first part of the paper contains some of the basic properties of graphs.
 The second part of the paper involves the determination of how many distinct graphs can be created using a specific number of matchsticks, which correspond to the edges. Since the number of nodes can vary, there is a table of (n = nodes, k = edges) values. A large number of figures are included. 

“Automorphic Numbers,” by Vernon deGuerre and R. A. Fairbairn
 An automorphic number is a number n such that the end of the square of n is equal to n. For example, 25 * 25 = 625, so 25 is automorphic. An analysis of the structure of automorphic numbers in base 10 as well as other bases and tables of all automorphic numbers up to 1000 digits in bases 6, 10 and 12 are included. 

“Division of Integers by Transposition,” by Charles W. Trigg
 There are numbers N such that if the leading digit d is transposed to the trailing end, the result is N / d. Methods for identifying such numbers and a list of some of them in various bases are given. 

“Tangrams,” by Harry Lindgren
 The set of seven pieces into which a square can be divided is known as the tans and the set of figures that they can be used to construct are called tangrams. These pieces have been around for some time and in this paper a method of numbering the tangrams for reference purposes is given.

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