Mathematical Association of America | 2012 | 312 páginas | rar - pdf | 2,3 Mb
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Martin Gardner enormously expanded the field of recreational mathematics with the Mathematical Games columns he wrote forScientific American for over 25 years and the more than 70 books he published. He also had a long relationship with the Mathematical Association of America, publishing articles in the MAA journals right up to his death in 2010. This book collects articles Gardner wrote for the MAA in the twenty-first century, together with other articles the MAA published from 1999 to 2012 that spring from and comment on his work. Martin Gardner's interests spanned geometry, number theory, graph theory, and probability, always communicated with engaging exposition often including games and puzzles. Eight works by Gardner himself, published between 1999 and 2010, are collected here and represent the breadth of his work, including his short fiction and lifelong interest in debunking pseudo-science. The remaining 33 chapters were written in response to Gardner's work and include several articles addressing open questions he posed. They come from The American Mathematical Monthly, Mathematics Magazine, The College Mathematics Journal, and Math Horizons and demonstrate how Gardner's influence continues beyond his columns for Scientific American. Although he took no mathematics in college, Martin Gardner inspired many mathematicians, professional and amateur, and his work was informed by frequent correspondence with other mathematics aficionados, both famous and unknown. He was even the basis for a character in a popular novel; his review of that work in included here. This book is a tribute to the deep and lasting impact of this prolific and brilliant writer. It is for anyone who, like Martin Gardner, loves mathematics.
Contents
Preface v
I Geometry 1
1 The Asymmetric Propeller 3
Martin Gardner
Gardner, paying tribute to dentist and geometer Leon Bankoff, discusses some of his unpublished results and concludes with an open question.
2 The Asymmetric Propeller Revisited 7
Gillian Saenz, Christopher Jackson, and Ryan Crumley
Three University of Texas students use dynamic geometry software to confirm Bankoff’s results and resolve Gardner’s question.
3 Bracing Regular Polygons As We Race into the Future 11
Greg W. Frederickson
A problem Gardner published in 1963 continues to spur generalizations and improved solutions around the world.
4 A Platonic Sextet for Strings 19
Karl Schaffer
The professor and dance company co-director details string polyhedra constructions for ten participants.
5 Prince Rupert’s Rectangles 25
Richard P. Jerrard and John E. Wetzel
A 17th century puzzle that Gardner posed in higher dimensions is here solved in the case of three-dimensional boxes.
II Number Theory and Graph Theory 35
6 Transcendentals and Early Birds 37
Martin Gardner
Gardnermoves from Liouville to an “innocent but totally useless amusement” that nonetheless captured the attention of Solomon Golomb.
7 Squaring, Cubing, and Cube Rooting 39
Arthur T. Benjamin
The professor and “mathemagician,” inspired as a high school student by Gardner’s tricks for mental calculations, extends some of them here.
8 Carryless ArithmeticMod 10 45
David Applegate, Marc LeBrun, and N. J. A. Sloane
Inspired by the carryless arithmetic of the game Nim, this trio of authors explores the number theory of a South Pacific island.
9 Mad Tea Party Cyclic Partitions 53
Robert Bekes, Jean Pedersen, and Bin Sha
Another playful trio analyzes cyclic arrangements that build from integer partitions in a Lewis Carroll setting.
10 The Continuing Saga of Snarks 65
sarah-marie belcastro
A type of graph, given a fanciful name by Gardner from Lewis Carroll, was the subject of a Branko Grunbaum conjecture for 39 years.
11 The Map-Coloring Game 73
Tomasz Bartnicki, Jaroslaw Grytczuk, H. A. Kierstead, and Xuding Zhu
Daltonism and half-dollar coins are used in this exploration of a Steven Brams game theory approach to the Four Color Theorem.
III Flexagons and Catalan Numbers 85
12 It’s Okay to Be Square If You’re a Flexagon 87
Ethan J. Berkove and Jeffrey P. Dumont
This article details the 1939 origin of flexagons at Princeton University and focuses on the neglected tetraflexagons.
13 The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons 103
Ionut E. Iacob, T. Bruce McLean, and Hua Wang
This trio of Georgia Southern University authors examines a once-illegal variety of flex and makes a connection between “pat classes” and Catalan numbers.
14 From Hexaflexagons to Edge Flexagons to Point Flexagons 109
Les Pook
An engineer and author of two books on flexagons considers the more general edge flexagons and recently discovered point flexagons.
15 Flexagons Lead to a Catalan Number Identity 113
David Callan
Examining the descent permutation statistic on flexagon pats leads the author to full binary trees and a combinatorial proof.
16 Convergence of a Catalan Series 119
Thomas Koshy and Z. Gao
Calculus is brought to bear on the infinite sum of Catalan number reciprocals and related series; and the golden ratio make appearances.
IV Making Things Fit 125
17 L-Tromino Tiling of Mutilated Chessboards 127
Martin Gardner
In his last MAA mathematics article, Gardner moves from classic chessboard domino tiling problems to new results.
18 Polyomino Dissections 135
Tiina Hohn and Andy Liu
The authors introduce a new technique for solving dissection problems, often presented in the context of quilts, leaving several puzzles for the reader.
19 Squaring the Plane 143
Frederick V. Henle and James M. Henle
A father and son team resolve Golomb’s “heterogenous tiling conjecture” and discuss another dozen open questions.
20 Magic Knight’s Tours 153
John Beasley
The author surveys results combining a knight’s tour on the chessboard with magic squares, including a computer-aided solution to a Gardner question.
21 Some New Results on Magic Hexagrams 159
Martin Gardner
Here Gardner focuses on three types of puzzles about placing numbers on six-pointed stars, mentioning a “rare mistake” of the British puzzlist Henry Dudeney.
22 Finding All Solutions to theMagic Hexagram 167
Alexander Karabegov and Jason Holland
The authors relate magic hexagrams to magic edge labelings of cubes, using card shuffling to enumerate distinct solutions.
23 Triangular Numbers, Gaussian Integers, and KenKen 173
John J. Watkins
Miyamoto’s contemporary puzzle is expanded to complex numbers where a different unique factorization adds to the challenge.
V Further Puzzles and Games 179
24 Cups and Downs 181
Ian Stewart
One of Gardner’s mathematical successors at Scientific American uses graph theory and linear algebra on two related puzzles.
25 30 Years of Bulgarian Solitaire 187
Brian Hopkins
Some recent math history explains this oddly-named puzzle on integer partitions, visualized with state diagrams and generalized to a new two-player game.
26 Congo Bongo 195
Hsin-Po Wang
A high school student uses state diagrams and Dennis Shasha’s detectives to open a tricky treasure chest.
27 Sam Loyd’s Courier Problem with Diophantus, Pythagoras,
and Martin Gardner 201
Owen O’Shea
A Classroom Capsule extends Gardner’s solution of related Sam Loyd puzzles to other army formations.
28 Retrolife and The Pawns Neighbors 207
Yossi Elran
An inverse version of Conway’s game Life, famously popularized by Gardner, is examined using chessboards.
29 RATWYT 213
Aviezri Fraenkel
The combinatorial game theorist uses the Calkin Wilf tree to devise a rational number version of Wythoff’s Nim.
VI Cards and Probability 219
30 Modeling Mathematics with Playing Cards 221
Martin Gardner
In addition to probability applications, Gardner uses a deck of cards for a discrete version of a fluid mixing puzzle and mentions a correction to W. W. Rouse Ball.
31 The Probability an Amazing Card Trick Is Dull 227
Christopher N. Swanson
Rook polynomials and the principle of inclusion-exclusion help determine the likelihood that the author’s students were underwhelmed.
32 The Monty Hall Problem, Reconsidered 231
Stephen Lucas, Jason Rosenhouse, and Andrew Schepler
These authors remind us of Gardner’s early role in this infamous problem that still “arouses the passions” and examine variations.
33 The Secretary Problem from the Applicant’s Point of View 243
Darren Glass
Changing perspective, the author reconsiders a classic strategy in order to help job seekers choose the best interview slot.
34 LakeWobegon Dice 249
Jorge Moraleda and David G. Stork
Lake Wobegon Dice, named after Garrison Keillor’s Minnesota town, have the property that each is “better than the set average.”
35 Martin Gardner’s Mistake 257
Tanya Khovanova
Another controversial problemabout probability and information is carefully discussed, putting Gardner in the company of Dudeney and Ball.
VII Other Aspects of Martin Gardner 263
36 Against the Odds 265
Martin Gardner
In this short story, a principal recognizes the potential in a student whose unconventional thinking irritates his teacher.
37 A ModularMiracle 271
John Stillwell
Gardner used an obscure result of Hermite and the limitations of 1970’s calculators for an April Fool’s Day prank.
38 The Golden Ratio—A Contrary Viewpoint 273
Clement Falbo
Building on a Gardner article in The Skeptical Inquirer, the author argues that “is not entirely astonishing.”
39 Review of The Mysterious Mr. Ammann by Marjorie Senechal 285
Philip Straffin
ThisMedia Highlight discusses an example of Gardner’s support of an amateurmathematician who independently discovered Penrose tiles.
40 Review of PopCo by Scarlett Thomas 287
Martin Gardner
This popular 2004 novel includes a character based on Gardner, so he was a natural choice to review the book.
41 Superstrings and Thelma 289
Martin Gardner
Gardner’s last MAA submission, a short story about a physics graduate student and a waitress who quips, “How are strings?”
Index 293
About the Editors 297
