Session 9: Mathematics – University of Copenhagen

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Board Game Studies Colloquium XX > Program > Session 9: Mathematics

Session 9: Mathematics

The Roots of Combinatorial Game Theory: History and Foresights

Sat 20 May, 10:30 - 11:00 (KUA3, Room 4A.0.69)

Dr Lisa Rougetet & Dr Abdallah Saffidine
Postdoctoral Researchers, University of Lille, France & University of New South Wales, Sydney

The mathematical subfield of Combinatorial Game Theory (CGT) studies how to predict the outcome of a game – win, loss, or draw – assuming optimal play from both players. The importance of CGT was firmly established with the publication of Winning Ways for your Mathematical Plays in 1982. In this two-volume text, Conway, Berlekamp, and Guy present a complete and deep theory, which can be deployed to analyze countless games. The central idea that enables such a fruitful treatment of games is the formal concept of disjunctive sum of games.

Our contribution is threefold. First, we define this "sum of games" and give examples of its use in a way that is accessible to non-mathematicians. Second, we recall the widely acknowledged timeline of CGT, which starts with the development of the Sprague-Grundy theory in the late 1930s. According to this timeline, before 1935, the mathematical study of games consisted largely of separate analyzes of extremely simple cases. Finally, we demonstrate for the first time that this perspective on the history of CGT is only partially accurate. To this end, we examine Emanuel Lasker's 1931 book on boardgames, Brettspiele der Völker, displaying the most important insights of the Sprague-Grundy theory.

Lisa Rougetet has a postdoctoral position at the University of Lille, France, where she teaches mathematics and works in the history of mathematics. She defended her PhD in 2014 on the history of combinatorial game theory. Her research interests include the history of the first Nim and chess playing machines and the history of combinatorial games programming in general and recreational mathematics. She is also concerned with connections between board games, computer algorithms and history, and their application in mathematics education. Her article "A Prehistory of Nim" was selected for the collection Best Writing on Mathematics 2015, Princeton University Press.

Abdallah Saffidine is a postdoctoral researcher at the University of New South Wales, Sydney, Australia where he works in the Artificial Intelligence and Algorithms groups. In 2015, Abdallah Saffidine was the recipient of a "Discovery Early-Career Research Award" by the Australian Research Council for the project Playing and Solving General Games. Abdallah's PhD thesis, Solving Games and All That, was elected "Best 2013 Dissertation" by the French Artificial Intelligence Association and "Best 2013 Publication" by the International Computer Games Association. Abdallah has a wide range of interests from games, planning, and other areas of decision-making to logic, complexity, and other areas of computer science.

Super Farmer: The First Board Game Using Twelve-Sided Dice

Sat 20 May, 11:00 - 11:30 (KUA3, Room 4A.0.69)

Michał Stajszczak
Games Historian & Game Developer, Warsaw, Poland

Dodecahedron was discovered by ancient Greeks but till the first half of the 20th century was known only to mathematicians. From five Platonic solids only a cube was widely used, for example as a random number generator in board games. Super Farmer is probably the first board game using twelve-sided dice. A history of this game and its mathematical model is described in this paper.

The game was first published in 1943 under the name Hodowla zwierzątek (A Little Animal Farm). The goal of the game is to collect at least one rabbit, one sheep, one pig, one cow and one horse. Players acquire animals throwing the dice. The main challenge during the designing of the game was how to obtain various probabilities for various animals. The six-sided dice did not give such possibility. But Karol Borsuk, the author of the game, was a professor of geometry, so he knew that a solid like the dodecahedron existed, and also knew how to make it of cardboard.

The game was self-made by the author and his wife and was sold in Warsaw till July 1944. Unfortunately, during the Warsaw Uprising in 1944 all copies were destroyed. After the war, professor Borsuk returned to scientific work at the University of Warsaw and the game was forgotten. 40 years later, after the death of the author, a single copy was found outside of Warsaw and was returned to Borsuk's family. Since 1997 the game has been produced under the name Super Farmer (with plastic dice, of course) and is now distributed in about 20 countries.

However, as it turned out, some of the ideas from the 1943 edition were criticized by 21st-century players. This paper explains the reasons for these critiques, and shows how the rules of the game were changed to meet the players' expectations.

Michał Stajszczak was born in Warsaw, Poland in 1956. In 1979 he obtained a Master of Science degree from Warsaw University of Technology (Faculty of Technical Physics and Applied Mathematics). He has been working as a co-owner of a wholesale company since 1990, selling board games, puzzles, cards, etc. He has designed rules for about ten board games and has translated the rules of more than one hundred board games into Polish for various editors, including Hasbro and Ravensburger. Between the years 2007-2013 he cooperated with Polish quarterly magazine Świat Gier Planszowych (World of Board Games), publishing texts about the history of games.

How dramatic is Snakes & Ladders?

Sat 20 May, 11:30 - 12:00 (KUA3, Room 4A.0.69)

Dr Jorge Nuno Silva (w/ Dr João Pedro Neto)
Assistant Professor of Mathematics, University of Lisbon, Portugal

What makes some abstract board games better than others, in the sense of providing a richer ludic experience? Four main parameters have been identified: depth, or  strategic complexity, is associated with the number of levels of play (games can be shallow as tic-tac-toe or deep as go); clarity tries to address the question of how easy it is, for the initiated player, to plan an attack or understand the dangers of a position (rithmomachia is opaque, chess is clear); decisiveness is the quality that allows substantial advantages to be turned into final victories in a natural way (in chess, if you are a queen ahead, you will win the game most of the time); finally, a game has drama if it is possible to overcome a difficult situation or balance a position by surprising strategic or tactical moves.

The authors tried to identify similar characteristics for some pure luck games. As they have shown in "Measuring Drama in Goose-like Games" (Board Game Studies Journal 10, 2016), drama can be extended to games of no skill in a natural way. They have now analysed some variants of Snakes & Ladders in a similar way and will share their results in this talk.

João Pedro Neto and Jorge Nuno Silva are professors at the University of Lisbon. Besides the joint paper cited in the abstract above, they authored the book Mathematical Games, Abstract Games published by Dover in 2013.