Mathematical Solitaires and Games

Benjamin L Schwartz

• Publisher: Routledge
• Year: 2019
• ISBN:
  9781351843065
• Edition: 1
• Topic: Mathematik
• Subtopic: Präkalkulation

A collection of solitaires and games which include sections on Solitiare Games like Knights Interchanges and The Stacked Playing Cards; Competitive games including SIM as a game of Chance and A winning Opening in Reverse Hex and also Solitaire games with toys like the Tower of Hanoi and Triangular Puzzle Peg.

SECTION TWO

Competitive Games

Since time beyond reckoning, people have competed against each other in games of skill. Tic-Tac-Toe and Dots-and-Squares are familiar examples of childhood games we have all played. Some of these pastimes are more profound and challenging than others. These harder games have ofttimes become the subject of study by mathematicians looking for a theory. Three such games are treated in this section, at different levels of completeness (because the amount known about them is different).

There is a general theory of competitive games, including games in which chance plays a factor (like backgammon) and games where pure skill is the only thing that matters (like chess). This theory was developed quite recently, as scientific discoveries go, by John Von Neumann, who presented the first organized approach in his 1945 book with Oskar Morgenstern The Theory of Games and Economic Behavior. Much of the material that would apply to the kind of games that recreational mathematicians play with came from earlier ideas by a writer named P. M. Grundy, whose key idea survives today in what is called the Grundy number of a position in a game. From the Grundy number you can tell whether, with proper play, the next player to move from that position can win. The game position must be represented by a point on a graph. This is an old idea in the literature of recreational mathematics. The first paper in this section gives a readable approach to graphs of games and Grundy numbers.

John Conway, a British mathematician, has very recently developed a different approach to competitive games, described in his book On Numbers and Games. It appears to be possibly more powerful than the classical graphical methods of Grundy and Von Neumann, but it is too early to be sure. At this time, no paper in JORM has attacked any game by Conway’s method, but we anticipate future developments along those lines in the near future.

In theory, with enough effort the graph of every competitive game can be constructed, and the Grundy numbers assigned. Therefore the winner is known for each playing position, and the winning move. In practice, this is only practical for the simplest games, like Tic-Tac-Toe. An individual game almost always requires a specific ingenious individual approach. In this section, we illustrate several such approaches.

One of these games, SIM, has been selected because it provides a veritable model of all of mathematical discovery. Here in the sequence of papers on SIM you will see this microunit of mathematical history unfold before your eyes in a period of about nine years, from its first appearance to its final solution. A problem emerged in the fertile brain of one individual. Then over a period of months or years other people looked at it and began to make discoveries. As more and more insights were obtained, a general outline began to emerge. After one or two tries fell short in one way or another, finally somebody put the pieces together right and the original problem was resolved. Most of the important papers along the way appeared in JORM, and we have brought them back here for your enjoyment.

Notice again the way this book gives a different view of mathematical recreations than others. If we were only concerned with telling you about the winning strategy for the second player in SIM, it would have sufficed to reprint only the last paper in the sequence by O’Brien. (Notice that the next-to-last, by Rounds and Yau, is wrongly titled. They don’t prove they actually have a winning strategy, only one that turns out very successfully in the tests they have given it.) But we have the opportunity to portray mathematics in its true character as a living, growing, very human activity with trials, hopes, frustrations, and sometimes triumph.

The fact that SIM is now “solved” does not by any means close the door to further study. A whole raft of additional questions still exist. In another paper in another journal (cited by O’Brien in his bibliography at the end of his paper) a different winning strategy was described. So the question naturally arises, how many different winning strategies are there? All known winning strategies require a computer for their execution because the choices become too complicated to manage mentally, or even with pencil and paper. Can someone find an approach that is practical for a player to remember and follow in the actual play of a game?

It is possible to commit suicide in SIM by completing a triangle when you have a different, safe move available. But suppose we consider strategies that may be less than optimal but are never suicidal. Can it ever occur that such a game is still forced to end before the last turn?You can see that SIM is still far from a closed book.

In the area of Dots-and-Squares, we again give you a sequence of papers describing a chain of discoveries. But in this case, the path of progress is far less complete, and the future more clouded. In Germany, the game is known as Käsekästchen, which means cheese boxes. Whatever the name, progress is at an early stage, despite the fact that this game has been around far longer than SIM. One of the interesting aspects of the theory that is developed in Ranucci’s paper is the fact that for even or odd n there are different formulas for the greatest number of moves before a square completion is forced.

The final paper in this section treats another novel and interesting game. It illustrates again how a familiar idea, like the game of HEX, can be given a new twist by reversing the rule defining winner and loser. Sometimes such an aberration leads to a new question that is easily answered, sometimes not. Here you have one sample of the latter case. Although reverse HEX is partially solved in Evans paper, a complete solution appears to be very difficult. This should be a fertile area for further research.

Compound Games with Counters

Cedric A.B. Smith

GAMES WITH ONE PILE OF COUNTERS

There is a well-known game played by two players (Abel and Baker) which runs as follows [1]: Let M be any given integer, greater than 1. There is a pile of cou...