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Poincare’s hyperbolic disk model.
Henri Poincar’e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved outward from the centre, everything got smaller in such a way that it would take an infinite time to reach the boundary.
The Tower of Hanoi is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:
Tower of Hanoi [image Wikimedia Commons].
John H Conway in 2009
[image Denise Applewhite, Princeton University].
John Horton Conway was a charismatic character, something of a performer, always entertaining his fellow-mathematicians with clever magic tricks, memory feats and brilliant mathematics. A Liverpudlian, interested from early childhood in mathematics, he studied at Gonville & Caius College in Cambridge, earning a BA in 1959. He obtained his PhD five years later, after which he was appointed Lecturer in Pure Mathematics.
In 1986, Conway moved to Princeton University, where he was Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics. He was awarded numerous honours during his career. Conway enjoyed emeritus status from 2013 until his death just two weeks ago on 11 April.
Video games generate worldwide annual sales of about $150 billion. With millions of people confined at home with time to spare, the current pandemic may benefit the industry. At the core of a video game is a computer program capable of simulating a range of phenomena in the real world or in a fantasy universe, of generating realistic imagery and of responding to the actions and reactions of the players. At every level, mathematics is crucial [TM184 or search for “thatsmaths” at irishtimes.com].
League of Legends, from Riot Games.
Continue reading ‘The Mathematics of Fair Play in Video Games’
Many problems in probability are solved by assuming independence of separate experiments. When we toss a coin, it is assumed that the outcome does not depend on the results of previous tosses. Similarly, each cast of a die is assumed to be independent of previous casts.
However, this assumption is frequently invalid. Draw a card from a shuffled deck and reveal it. Then place it on the bottom and draw another card. The odds have changed: if the first card was an ace, the chances that the second is also an ace have diminished.
Alice and Bob, are enjoying a drink together. Sitting in a bar-room, they take turns placing beer mats on the table. The only rules of the game are that the mats must not overlap or overhang the edge of the table. The winner is the player who puts down the final mat. Is there a winning strategy for Alice or for Bob?
Image from Flickr.
We start with the simple case of a circular table and circular mats. In this case, there is a winning strategy for the first player. Before reading on, can you see what it is?
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Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player’s coin lands completely within a square, he or she wins a coin of equal value. If the coin crosses a dividing line, it is lost.
The playing board for Franc-Carreau is shown above, together with a winning coin (red) contained within a square and a loosing one (blue) crossing a line. As the precise translation of franc-carreau appears uncertain, the name “fair square” would seem appropriate.
The question is: What size should the coin be to ensure a 50% chance of winning?
How many ways can a deck of cards be arranged? It is very easy to calculate the answer, but very difficult to grasp its significance.
Hex is an amusing game for two players, using a board or sheet of paper divided into hexagonal cells like a honeycomb. The playing board is rhomboidal in shape with an equal number of hexagons along each edge. Players take turns placing a counter or stone on a single cell of the board. One uses white stones, the other black. Or red and blue markers can be used on a paper board.
11 x 11 Hex Board. Image from iggamecenter.com
Continue reading ‘Fun and Games on a Honeycombed Rhomboard.’
Game theory deals with mathematical models of situations involving conflict, cooperation and competition. Such situations are central in the social and behavioural sciences. Game Theory is a framework for making rational decisions in many fields: economics, political science, psychology, computer science and biology. It is also used in industry, for decisions on manufacturing, distribution, consumption, pricing, salaries, etc.
Theory of Games and Economic Behavior.
Centre: John von Neumann. Right: Oskar Morgenstern.
During the Cold War, Game Theory was the basis for many decisions concerning nuclear strategy that affected the well-being of the entire human race.
John Nash, who was the subject of the book and film A Beautiful Mind, won the Abel Prize recently. But his journey home from the award ceremony in Norway ended in tragedy [see this week’s That’s Maths column (TM069): search for “thatsmaths” at irishtimes.com].
Russell Crowe as John Nash in the movie A Beautiful Mind.
Have you ever tried to build a high stack of coins? In theory it’s fine: as long as the centre of mass of the coins above each level remains over the next coin, the stack should stand. But as the height grows, it becomes increasingly trickier to avoid collapse.
Ten chocolate gold grain biscuits, with a hangover of about one diameter.
In theory it is possible to achieve an arbitrarily large hangover — most students find this out for themselves! In practice, at more than about one coin diameter it starts to become difficult to maintain balance.
Continue reading ‘Biscuits, Books, Coins and Cards: Massive Hangovers’
Long ago in the Gupta Empire, a great-but-greedy mathematician, Grababundel, presented to the Maharaja a new game that he had devised, called Chaturanga.
Thirty-two of the Maharaja’s subjects, sixteen dressed in white and sixteen in black, were assembled on a field divided into 64 squares. There were rajas and ranis, mahouts and magi, fortiers and foot-soldiers. Continue reading ‘Chess Harmony’
What is the most beautiful rectangular shape? What is the ratio of width to height that is most aesthetically pleasing? This question has been considered by art-lovers for centuries and one value appears consistently, called the golden ratio or Divine proportion. Continue reading ‘The Beautiful Game’
Whether or not you enjoy solving them, those 9×9 grids with numbers and blank cells cannot have escaped your notice. Sudoku puzzles have swept the world since exploding on the scene in 2005. They are found in newspapers everywhere, providing daily amusement to all who like a minor mathematical challenge. Continue reading ‘No Maths Involved!’
Sprouts is a simple and delightfully subtle pencil-and-paper game for two players. The game is set up by marking a number of spots on a page. Each player makes a move by drawing a curve that joins two spots, or that loops from a spot back to itself, without crossing any lines drawn earlier, and then marking a new spot on the curve. Continue reading ‘Sproutology’
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