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RMM Number1 March 2014 Hi Edition Associa¸c˜ao Ludus, Museu Nacional de Hist´oria Natural e da Ciˆencia, Rua da Escola Polit´ecnica 56 1250-102 Lisboa, Portugal email: [email protected] URL: http://rmm.ludus-opuscula.org Managing Editor Carlos Pereira dos Santos, [email protected] Center for Linear Structures and Combinatorics, University of Lisbon Editorial Board Aviezri S. Fraenkel, [email protected], Weizmann Institute Carlos Pereira dos Santos Colin Wright, [email protected], Liverpool Mathematical Society David Singmaster, [email protected], London South Bank University David Wolfe, [email protected], Gustavus Adolphus College Edward Pegg, [email protected], Wolfram Research Jo˜ao Pedro Neto, [email protected], University of Lisbon Jorge Nuno Silva, [email protected], University of Lisbon Keith Devlin, [email protected], Stanford University Richard Nowakowski, [email protected], Dalhousie University Robin Wilson, [email protected], Open University Thane Plambeck, [email protected], Counterwave, inc Informations The Recreational Mathematics Magazine is electronic and semiannual. The issues are published in the exact moments of the equinox. The magazine has the following sections (not mandatory in all issues): Articles Games and Puzzles Problems MathMagic Mathematics and Arts Math and Fun with Algorithms Reviews News Contents Page Editorial .................................... 3 Problems: Colin Wright The Mutilated Chess Board (Revisited) ............. 4 Games and Puzzles: David Singmaster Vanishing Area Puzzles ........................ 10 Articles: Robin Wilson Lewis Carroll in Numberland .................... 22 Articles: Alda Carvalho, Carlos Santos, Jorge Nuno Silva Mathematics of Soccer ........................ 35 Articles: Thane Plambeck, Greg Whitehead The Secrets of Notakto: Winning at X-only Tic-Tac-Toe . 49 Articles: Svenja Huntemann, Richard J. Nowakowski Doppelganger¨ Placement Games .................. 55 Editorial The Recreational Mathematics Magazine is published by the Ludus Asso- ciation with the support of the Interuniversity Centre for the History of Science and Technology. The journal is electronic and semiannual, and fo- cuses on results that provide amusing, witty but nonetheless original and scientifically profound mathematical nuggets. The issues are published in the exact moments of the equinox. It’s a magazine for mathematicians, and other mathematics lovers, who enjoy imaginative ideas, non-standard ap- proaches, and surprising procedures. Occasionally, some papers that do not require any mathematical background will be published. Examples of topics to be addressed include: games and puzzles, problems, mathmagic, mathematics and arts, history of mathematics, math and fun with algorithms, reviews and news. Recreational mathematics focuses on insight, imagination and beauty. His- torically, some areas of mathematics are strongly linked to recreational mathematics - probability, graph theory, number theory, etc. Thus, re- creational mathematics can also be very serious. Many professional mathe- maticians confessed that their love for math was gained when reading the articles by Martin Gardner in Scientific American. While there are conferences related to the subject as the amazing Gathering for Gardner, and high-quality magazines that accept recreational papers as the American Mathematical Monthly, the number of initiatives related to this important subject is not large. This context led us to launch this magazine. We will seek to provide a qua- lity publication. Ideas are at the core of mathematics, therefore, we will try to bring in focus amazing mathematical ideas. We seek sophistication, imagination and awe. Lisbon, 1st March, 2014, Ludus Association Interuniversity Centre for the History of Science and Technology Problems The Mutilated Chess Board (Revisited) Colin Wright Liverpool Mathematical Society [email protected]; @ColinTheMathmo; www.solipsys.co.uk Abstract: The problem of the Mutilated Chessboard is much loved and familiar to most of us. It’s a wonderful example of an impossibility proof, and at the same time completely accessible to people of all ages and experience. In this work we will delve a little deeper, find an unexpected connection, and discover the true power of abstract mathematics while still seeing the concrete applica- tions in action. Key-words: impossibility proofs, Marriage Theorem, matching problems. Puzzle enthusiasts know that a really good puzzle is more than just a problem to solve. The very best problems and puzzles can provide insights that go beyond the original setting. Sometimes even classic puzzles can turn up something new and interesting. Some time ago while re-reading one of Martin Gardner’s books [1] I happened across one of these classics, the problem of the Mutilated Chess Board, and was surprised to learn something new from it. That’s what I want to share with you here. The way I present it here is slightly unusual - bear with me for a moment. So consider a chess board, and a set of dominos, each of which can cover exactly two squares. It’s easy to cover the chess board completely and exactly with the dominos, and it will (rather obviously) require exactly 32 dominos to do so. We might wonder in how many ways it can be covered, but that’s not where I’m going. Figure 1: Standard Chessboard. Recreational Mathematics Magazine, Number 1, pp. 4–9 Colin Wright 5 If we mutilate the chess board by removing one corner square (or any single square, for that matter) then it’s clearly impossible to cover it exactly, because now it would now take 31, 5 dominos. Something not often mentioned is that cutting off two adjacent corners leaves the remaining squares coverable, and it takes 31 dominos. It’s not hard - pretty much the first attempt will succeed. The classic problem then asks - is it possible to cover the board when two op- posite corners are removed? The first attempt fails, and the second, and the third, and after a while you start to wonder if it’s possible at all. One of the key characteristics of mathematicians and puzzlers is that they don’t simply give up, they try to prove that it’s impossible. Insight strikes when (if!) you realise that every attempt leaves two squares un- covered, and they are the same colour. Why is this important? The reason is simple. Each domino must cover exactly one black and one white, and the two squares we’ve removed are the same colour. As we cover pairs of squares, we must cover the same number of blacks as whites. If the number of blacks and whites is unequal, it can’t be done. Figure 2: Mutilated Chessboard. It’s a wonderful example of being able to show that something is impossible, without having to examine all possible arrangements. And that’s where the discussion usually stops. Some people look for generali- sations in different ways, and that has led to some interesting extensions of the ideas, but again, that’s not where I’m going. Clearly in order to cover the mu- tilated board it’s necessary that there be the same number of blacks as whites. Is this sufficient? Let’s start with the simple cases. If you remove any single black and single white and try to cover the remain- der it seems always to be possible. Indeed, in 1973 mathematician Ralph E. Gomory [3] showed with a beautifully simple and elegant argument that it is Recreational Mathematics Magazine, Number 1, pp. 4–9 6 The Mutilated Chess Board (Revisited) always possible to cover the board if just one black and one white square are removed. So that’s a good start. What about if two whites and two blacks are removed? Clearly if one corner gets detached by the process then it’s no longer coverable, but let’s suppose the chess board remains connected. Can it always be covered? Yes it can, although I’ll leave the details to the interested reader. I’ve yet to find a truly elegant argument (and would be interested in seeing one. Please.) And so we continue. What if three whites and three blacks are removed (still leaving the board connected.) Can it then always be covered? No. Oh. Well, that was quick. Challenge: Find an example of a chess board with three whites and three blacks removed, still connected, but can’t be covered. There is a hint later [X]. So consider. Clearly it’s necessary that the number of blacks is the same as the number of whites, but equally clearly it’s not sufficient. So what condition is both necessary and sufficient? Specifically: Given a chess board with some squares removed (but still connected), can we show that it’s not possible to cover it with dominos, without having to try every possible arrangement? Given that this is a paper for the Recreational Mathematics Magazine it won’t surprise you that the answer is yes. It’s an elegant and slightly surprising re- sult concerning matchings between collections, and goes by the name of “Hall’s Theorem”, or sometimes “Hall’s Marriage Theorem” [2]. I’ll explain it briefly here, and then show an unexpected application [Z]. The original setting is this. Suppose we have a collection of men and a collection of women, and each woman is acquainted with some of the men. Our challenge is to marry them all off so that each woman is only married to a man she knows. Under what circumstances is this possible? In another formulation, suppose we have a collection of food critics. Naturally enough, they all hate each other, and refuse to be in the same room as each other. Also unsurprisingly, each has restaurants that they won’t ever eat at again. Is it possible for them all to eat at a restaurant they find acceptable, but without having to meet each other? These are “Matching” problems - we want to match each item in one collec- tion to an item in the other.
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