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Invited Paper Session Abstracts - The Serious Side of Recreational Mathematics

Friday, August 2, 1:30 p.m. - 3:50 p.m., Duke Energy Convention Center, Room 200

Recreational mathematics covers a wide variety of themes: card tricks, board games, puzzles, origami, and art are just a few. The use of “recreational” gives the impression that research in these topics is more of a pastime than an investigation with depth to it. However, when you look below the surface, there is a surprising amount of complexity to the subjects being studied. This invited paper session will include experts in the many topics in recreational math showing how starting with a fun puzzle, game, or story can take one on a trip to deep mathematics.

Our goal is to show the myriad of topics gathered underneath the recreational umbrella rather than highlight one topic. The gamut of this runs from the combinatorial questions in designing unique puzzles and using origami in designing robots, to the powerful logic in Knights and Knaves puzzles, the geometric structures hidden in the game SET, and using number theory to create new tricks with playing cards.

Robert Vallin, Lamar University

Sponsor: SIGMAA on Recreational Mathematics

Bingo Paradoxes

1:30 p.m. - 1:50 p.m.
Art Benjamin, Harvey Mudd College


Suppose you walk by a large Bingo Parlor and hear someone excitedly shout Bingo! Is it more likely that the Bingo is Horizontal or Vertical? Or maybe it's Diagonal? The answer to this, and related questions, will surprise you.


Garden of Eden Partitions for Bulgarian and Austrian Solitaire

2:00 p.m. - 2:20 p.m.
James Sellers, Penn State University


In the early 1980s, Martin Gardner popularized the game called Bulgarian Solitaire through his writings in Scientific American. After a brief introduction to the game, we will discuss a few results proven about Bulgarian Solitaire around the time of the appearance of Gardner's article and then quickly turn to the question of finding an exact formula for the number of Garden of Eden partitions that arise in this game. I will then introduce a related game known as Austrian Solitaire and consider a similar question about the Garden of Eden states that appear. The talk will be completely self-contained and should be accessible to a wide-ranging audience, including undergraduate students and faculty members. This is joint work with Brian Hopkins and Robson da Silva.


Geometry, Combinatorics and the Game of SET

2:30 p.m. - 2:50 p.m.
Liz McMahon, Lafayette College


The deck of cards for the game of SET is an excellent model for the finite affine geometry \(AG(4,3)\) and provides an entry to surprisingly beautiful combinatorial structure theorems for that geometry. In this talk, we’ll explore that geometry and find a beautiful structure hiding within the collections of cards that have no sets (called maximal caps). There’s a connection to the outer automorphisms of \(S_6\) as well, but you don’t need to know what those are to follow what we’ll do.


Throwing Together a Proof of Worpitzky's Identity

3:00 p.m. - 3:20 p.m.
Steve Butler, Iowa State University


Juggling is about patterns and we can use mathematics to explore and count the different patterns that emerge. By combining together different ways to describe juggling patterns we will give a bijective proof of Worpitzky's identity. We will also discuss the odd juggler problem (and it is not that all jugglers are odd).


Domino Variations

3:30 p.m. - 3:50 p.m.
Bob Bosch, Oberlin College


Ken Knowlton was the first to devise an algorithm for arranging complete sets of dominos into mosaics that resemble user-supplied target images. Knowlton's method is a two-step approach that begins by partitioning the "canvas" into domino-sized "slots" and concludes by assigning actual numbered dominos to these slots. Donald Knuth pointed out that each step of Knowlton's method is an instance of the linear assignment problem (which is solvable in polynomial time). Bosch's integer programming model is more computationally intensive. It tackles both steps at the same time. In this talk, we compare the Knowlton/Knuth and Bosch methods, we discuss the use of non-rectangular canvases, and we present a method for constructing a domino mosaic that contains a hidden second image.