A relation between Catalan Numbers and World Series type problems

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This issue of *The College Mathematics Journal* is devoted to the mathematics of Martin Gardner (1914-2010), prolific writer on mathematics and science, best known for the immensely influential series of “Mathematical Games” columns that appeared in *Scientific American* from December of 1956 through 1981.

Appropriate to the memory Martin Gardner, the issue includes a numerical challenge, several puzzles and drawings, and a crossword.

**Hexaflexagons ** (read the full article)

*Martin Gardner *

A reprint of Martin Gardner’s very first “Mathematical Games” column, which appeared December 1956 in *Scientific American*. Here Gardner introduces flexagons (paper polygons folded from straight or crooked strips of paper which have the fascinating property of changing their faces when they are “flexed”), discusses their history prior to World War II, and explains how to make several hexaflexagons. The reprint, part of a special issue of *The College Mathematics Journal* devoted to “Martin Gardner’s Mathematics,” is followed by two contemporary papers that describe some of the ways that the study of flexagons has developed since 1956. Susan Goldstine and Ethan Berkhove provided vital assistance editing all three flexagon papers.

**The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons** (read the full article)

*Ionut E. Iacob, T. Bruce McLean, and Hua Wang *

Regular hexaflexagons mysteriously change faces as you pinch flex them. This paper describes a different flex, the V-flex, which allows the hexahexaflexagon (with only 9 faces under the pinch flex) to have 3420 faces. The article goes on to explain the classification of triangle orientations in a hexaflexagon and gives an example of the combinatorics of flexagons.

**From Hexaflexagons to Edge Flexagons to Point Flexagons **(read the full article)

*Les Pook *

Flexagons, introduced to a wide audience 50 years ago by Martin Gardner, now have an extensive literature and are an active field of research. This paper describes two kinds: edge flexagons and point flexagons, and gives an example of one means of classification.

**Cups and Downs **(read the full article)

*Ian Stewart*

Martin Gardner wrote about a coin-flipping trick, performed by a blindfolded magician. The paper analyses this trick, and compares it with a similar trick using three cups flipped in pairs. Several different methods of analysis are discussed, including a graphical analysis of the state space and a representation in terms of a matrix. These methods can also solve a more general problem about inverting n cups m at a time, whose answer is surprisingly complicated.

**Martin Gardner’s Mistake** (read the full article)

*Tanya Khovanova *

When Martin Gardner first presented the Two-Children problem, he made a mistake in its solution. Later he corrected the error, but unfortunately the incorrect solution is more widely known than his correction. In fact, a Tuesday-Child variation of this problem went viral in 2010, and the same flaw keeps reappearing in proposed solutions of that problem too. In this article, we re-visit Martin Gardner’s correction and discuss the new problem in detail.

**Mad Tea Party Cyclic Partitions** (read the full article)

*Robert Bekes, Jean Pedersen, and Bin Shao*

Martin Gardner’s The Annotated Alice, and Robin Wilson’s Lewis Carroll in Numberland led the authors to put this article in a fantasy setting. Alice, the March Hare, the Hatter, and the Dormouse describe a straightforward, elementary algorithm for counting the number of ways to fit n identical objects into k cups arranged in a circle. The authors call these cyclic arrangements Mad Tea Party (MTP) partitions. Our algorithm enumerates MTP partitions (in sets of odd and even entries), then, by ordering them and striking out duplicates, one obtains the ordinary partitions for any positive integer n.

**Triangular Numbers, Gaussian Integers, and KenKen** (read the full article)

*John J. Watkins*

Latin squares form the basis for the recreational puzzles sudoku and KenKen. In this article we show how useful several ideas from number theory are in solving a KenKen puzzle. For example, the simple notion of triangular number is surprisingly effective. We also introduce a variation of KenKen that uses the Gaussian integers in order to illustrate the concept of unique factorization.

**Carryless Arithmetic Mod 10** (read the full article)

David Applegate, Marc LeBrun, and N. J. A. Sloane

What might arithmetic look like on an island that eschews carry digits? How would primes, squares and other number theoretical concepts play out on such an island?

**Bracing Regular Polygons As We Race into the Future** (read the full article)

*Greg N. Frederickson *

How many rods does it take to brace a square in the plane? Once Martin Gardner’s network of readers got their hands on it, it turned out to be fewer than Raphael Robinson, who originally posed the problem, thought. And who could have predicted the stunning solutions found subsequently for various generalizations of the problem?

**Squaring, Cubing, and Cube Rooting** (read the full article)

*Arthur T. Benjamin *

We present mentally efficient algorithms for mentally squaring and cubing 2-digit and 3-digit numbers and for finding cube roots of numbers with 2-digit or 3-digit answers.

**A Platonic Sextet for Strings** (read the full article)

*Karl Schaffer*

The use of traditional string figures by the Dr. Schaffer and Mr. Stern Dance Ensemble led to experimentation with polyhedral string constructions. This article presents a series of polyhedra made with six loops of three colors which sequence through all the Platonic Solids.

**The Play’s the Thing!** (read the full article)

*Gary Kennedy and Stephen Kennedy*

A crossword puzzle.

**Magic Knight’s Tours** (read the full article)

*John D. Beasley *

The topic of the magic knight’s tour, discussed by Martin Gardner in one of his books, is here brought up to date in the light of modern computer discoveries.

**The Secretary Problem from the Applicant’s Point of View** (read the full article)

*Darren Glass*

A 1960 “Mathematical Games” column describes the problem, now known as the Secretary Problem, which asks how someone interviewing candidates for a position should maximize the chance of hiring the best applicant. This note looks at how an applicant should respond, if they know the interviewer uses this optimal strategy. We show that all but the very top applicants have the best chance of being hired if they arrange to be the last person interviewed.

**The Continuing Saga of Snarks** (read the full article)

*sarah-marie belcastro *

We review the history of snarks and give a selected survey of recent research. The article and snarks themselves are much more interesting than this summary makes them sound.

**Polyomino Dissections** (read the full article)

*Tiina Hohn and Andy Liu *

One of Gardner’s passions was to introduce puzzles into the classroom. From this point of view, polyomino dissections are an excellent topic. They require little background, provide training in geometric visualization, and mostly they are fun. In this article, we put together a large collection of such puzzles, introduce a new approach in solving them, and discuss the practical considerations in presenting these puzzles to children.

**PROBLEMS AND SOLUTIONS** (read the full article)

**MEDIA HIGHLIGHTS** (read the full article)