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
Abstract
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
Abstract
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
Abstract
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
Colegiul National Fratii Buzeşti
Craiova, România
Florentin Smarandache
Universitatea, New Mexico
smarand@unm.edu
Universitatea, New Mexico
smarand@unm.edu
Abstract
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
Samantha Fairchild
Houghton College
samantha.fairchild15@houghton.edu
Houghton College
samantha.fairchild15@houghton.edu
Viorel Nitica
West Chester University
vnitica@wcupa.edu
West Chester University
vnitica@wcupa.edu
Samuel Simon
Carnegie Mellon University
slsimon@cmu.edu
Carnegie Mellon University
slsimon@cmu.edu
Abstract
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 n
and 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.
https://www.amazon.com/Topics-Recreational-Mathematics-2-2016/dp/1534964843/
and 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. https://www.amazon.com/Topics-Recreational-Mathematics-2-2016/dp/1534964843/
Congratulations for your continuous good work.
ReplyDelete