Abstracts
of the papers that appeared in “Journal of Recreational Mathematics, Volume 1,
Number 2, 1968"
Abstracts were not included with the original papers,
so all of the abstracts were written by Charles Ashbacher.
“Compound Games With Counters,” by Cedric A. B. Smith
Abstract
Many different games can be defined where there is one or more pile of counters and two players alternate removing a nonzero number of counters. The general rule is that the number of counters removed must be less than a fixed number M. Winners of games of this type can either be declared the winner or loser if they remove the last counter.
This paper contains an analysis of several games of this type and some of them are Nim or a variation. When there are multiple piles, the games can be played in either a conjunctive or disjunctive mode. In a conjunctive mode, the players must remove at least one counter from every pile in a move and in a disjunctive mode, counters are removed from only one pile in a move. Another variant has the player making a move in at least one pile but not in all.
Many different games can be defined where there is one or more pile of counters and two players alternate removing a nonzero number of counters. The general rule is that the number of counters removed must be less than a fixed number M. Winners of games of this type can either be declared the winner or loser if they remove the last counter.
This paper contains an analysis of several games of this type and some of them are Nim or a variation. When there are multiple piles, the games can be played in either a conjunctive or disjunctive mode. In a conjunctive mode, the players must remove at least one counter from every pile in a move and in a disjunctive mode, counters are removed from only one pile in a move. Another variant has the player making a move in at least one pile but not in all.
“Curiosa on 1968,” by Leon Bankoff
Abstract
This short paper contains fifteen ways of expressing 1968 cubed as the sum of three different cubes of positive integers as well as two basic identities using the Fibonacci numbers.
This short paper contains fifteen ways of expressing 1968 cubed as the sum of three different cubes of positive integers as well as two basic identities using the Fibonacci numbers.
“Some Approximate Dissections,” by Harry Lindgren
Abstract
In this paper, polygons are dissected into two or more pieces, where the pieces are very close to identical. For example, in the first dissection, the two pieces are in the ratio of 99:101. The figure dissected is in nearly all cases the regular pentagon. The pieces that it is dissected into are the equilateral triangle, square, the regular hexagon and Greek and Latin crosses. In nearly all cases the error is so small as to be capable of fooling a viewer if drawn correctly. The non-pentagonal figures dissected are the hexagon and octagon.
In this paper, polygons are dissected into two or more pieces, where the pieces are very close to identical. For example, in the first dissection, the two pieces are in the ratio of 99:101. The figure dissected is in nearly all cases the regular pentagon. The pieces that it is dissected into are the equilateral triangle, square, the regular hexagon and Greek and Latin crosses. In nearly all cases the error is so small as to be capable of fooling a viewer if drawn correctly. The non-pentagonal figures dissected are the hexagon and octagon.
“Patterns in Primes,” by Leslie E. Card
Abstract
There are many different patterns that one can find in the prime numbers. In this case, prime numbers that are reversible, cyclic and can be used to create squares and cubes are presented. Another interesting operation is to start with a 5-digit prime and add digits one at a time to form a continuous series of overlapping 5-digit primes.
There are many different patterns that one can find in the prime numbers. In this case, prime numbers that are reversible, cyclic and can be used to create squares and cubes are presented. Another interesting operation is to start with a 5-digit prime and add digits one at a time to form a continuous series of overlapping 5-digit primes.
“A Prime Number Sieve,” by Douglas A. Engel
Abstract
Lines with slopes of 1/1, 1/2, 1/3, . . . starting at (0,0) are drawn in the first quadrant of the coordinate plane. Drawing vertical lines down from the lattice points on the line with slope 1/1 and checking for points of intersection of the other lines will reveal primes and composites.
Lines with slopes of 1/1, 1/2, 1/3, . . . starting at (0,0) are drawn in the first quadrant of the coordinate plane. Drawing vertical lines down from the lattice points on the line with slope 1/1 and checking for points of intersection of the other lines will reveal primes and composites.
“A Digital Bracelet for 1968,” by Charles W. Trigg
Abstract
A digital bracelet is constructed by starting with four digits, computing the unit digit of their sum, affixing that to the end, then computing the unit digit of the sum of the preceding four digits and then repeating the process. Given the limited number of four digit numbers, the operation will eventually repeat and that cycle for starting with 1968 is given. Other features pointed out are the patterns that emerge when the elements of the cycle are placed in rows in a certain way.
A digital bracelet is constructed by starting with four digits, computing the unit digit of their sum, affixing that to the end, then computing the unit digit of the sum of the preceding four digits and then repeating the process. Given the limited number of four digit numbers, the operation will eventually repeat and that cycle for starting with 1968 is given. Other features pointed out are the patterns that emerge when the elements of the cycle are placed in rows in a certain way.
“’Hit-and-Run’ on a Graph,” by Jorg Nievergelt and
Steve Chase
Abstract
Shannon’s switching game begins with a graph with two distinguished nodes. There are two players that are called “Cut” and “Short” respectively. For each play, Cut deletes a node connection while Short claims a connection as immune from cutting. Short wins if a path between the two distinguished nodes is preserved at the end while Cut wins if there is no such path.
Shannon’s switching game begins with a graph with two distinguished nodes. There are two players that are called “Cut” and “Short” respectively. For each play, Cut deletes a node connection while Short claims a connection as immune from cutting. Short wins if a path between the two distinguished nodes is preserved at the end while Cut wins if there is no such path.
Hit-and-Run is
a similar game played on a 4 × 4 board by two players called H and V. The goal
of H is to add node connections so that there is a horizontal path between two
nodes on the vertical sides and the goal of V is to create a vertical path
between the nodes on the horizontal sides. This paper is a report on a computer
program that was written to play the game and insights that were generated.
“A 1968 Magic Square Composed of Leap Years,” by Leon
Bankoff
Abstract
This is a four-by-four magic square composed of leap years where the magic sum is 1968.
This is a four-by-four magic square composed of leap years where the magic sum is 1968.
“Integers of the Form N^3 + M^5,” by Charles W. Trigg
Abstract
This is a complete list of all integers P < 10,000 which are the sum of positive cube and positive fifth powers.
This is a complete list of all integers P < 10,000 which are the sum of positive cube and positive fifth powers.
“A Special Square Array of the Nine Digits,” by
Charles W. Trigg
Abstract
The digits one through 9 are put into a three-by-three array that has a set of interesting properties when rotated.
The digits one through 9 are put into a three-by-three array that has a set of interesting properties when rotated.
“Square Solitaire and Variations,” by Donald C. Cross
Abstract
Square solitaire is a game where fifteen pegs are placed on a four-by-four board with sixteen holes. A move is when one peg jumps another to an empty space and the jumped peg is removed. The goal is to complete a set of moves so that only one peg remains.
Solutions to the four-by-four game are given as well as a solution for a five-by-five, a seven-by-seven and an eight-by-eight game.
Square solitaire is a game where fifteen pegs are placed on a four-by-four board with sixteen holes. A move is when one peg jumps another to an empty space and the jumped peg is removed. The goal is to complete a set of moves so that only one peg remains.
Solutions to the four-by-four game are given as well as a solution for a five-by-five, a seven-by-seven and an eight-by-eight game.
“A Fundamental Dissection Puzzle,” by Kobon Fujimura
Abstract
The dissections examined in this paper are those of a square board that is segmented into squares of equal size. All dissection lines must follow the borders of squares. Four-by-four and five-by-five boards are dissected into identical parts and when there is an odd number of squares, the central one is not considered part of the dissection.
The dissections examined in this paper are those of a square board that is segmented into squares of equal size. All dissection lines must follow the borders of squares. Four-by-four and five-by-five boards are dissected into identical parts and when there is an odd number of squares, the central one is not considered part of the dissection.
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