Wednesday, July 6, 2016

List of abstracts of the papers that appeared in "Topics in Recreational Mathematics Volume 7" edited by Charles Ashbacher

List of abstracts of the papers that appeared in Topics in Recreational Mathematics Volume 7 edited by Charles Ashbacher, ISBN 978-1534964846

Is There Safety in Numbers? A Study of California’s Sanctuary Cities

Paul M. Sommers
Economics Department
Middlebury College


 The author examines eight different crime rates (three defined for crimes against persons, five for crimes against property) in sanctuary cities to the corresponding average of all other cities in California of comparable size.  The results indicate that crime rates in 2013 were significantly higher in all sanctuary cities and, in particular, for California’s fifteen cities with a population over 75,000, four crime rates ─ robbery, motor vehicle theft, aggravated assault, and murder ─ were significantly higher in sanctuary cities.

Some Notes on the Evaluation of Magic Sums of
Magic Squares

Hossein Behforooz

 We have a formula to calculate the magic sum of magic squares with positive consecutive integers.  But we do not have a formula when entries of the magic squares are not consecutive positive integers.  In this paper, we have some comments about this situation.

Making the Grid Square-Free

Dr. Moloy De


Considered is a finite regular grid of squares. The main question addressed is on the minimum number of edges needed to be removed to make the grid square-free.

Keywords: Lattice Grid, Matchstick Puzzle.

About the Second Droz-Farny Circle

Ion Pătraşcu
Colegiul National Fratii Buzeşti
Craiova, România
Florentin Smarandache
Universitatea, New Mexico
 In this article, we prove the theorem relative to the second Droz-Farny circle, and a sentence that generalizes it.

Tilings of Quadrants by L-ominoes and Notched Rectangles
Aaron Calderon
University of Nebraska-Lincoln                         
Samantha Fairchild
Houghton College
Viorel Nitica
West Chester University
Samuel Simon
Carnegie Mellon University
  In this paper, we examine tilings of the four quadrants in a Cartesian coordinate system by tile sets consisting of L-shaped polyominoes and notched rectangles. We start with tile sets consisting of an L-shaped polyomino and a notched rectangle, appearing from the dissection of a n × n, n ≥ 3, square, and of the symmetries of these two tiles about the first diagonal. In this case, a tiling of a quadrant is said to follow the rectangular pattern if it reduces to a tiling by n × n  squares, each of the squares in turn tiled by an L-shaped polyomino and a notched rectangle. We show that for every tile sets as above, with the possible exception of one appearing from the dissection of a 3 × 3 square, there exists at least a quadrant for which any tiling has to follow the rectangular pattern. We further consider tilings of the quadrants with tile sets appearing from similar dissections of mn × n, n ≥ 3, m ≥ 2, rectangles and show that for every one of them there exists at least a quadrant for which every tiling has to follow the rectangular pattern. Our results have consequences for tilings of other regions in plane, in particular rectangles and infinite half strips. A rectangle can be tiled by the above tile sets if and only if both sides are divisible by nand one side is divisible by mn, and an infinite half strip can be tiled if and only if its width is divisible by n. In all of the above cases, the rectangular pattern of a tiling persists if we add an extra n × n square to the tile set. Our technique of proof is to use induction along a staircase line built out of n × n squares and to show that the existence of a tile in an irregular position propagates towards the edges of the quadrant, and eventually leads to a contradiction. Further, we look at similar tile sets appearing from dissections of rectangles m × n of coprime sides  m, n n ≥ 2. Here there exist tilings that do not follow the rectangular pattern. 

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