Posts Tagged 'Games'

Franc-carreau or Fair-square

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?

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Factorial 52: A Stirling Problem

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.


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Fun and Games on a Honeycombed Rhomboard.

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

11 x 11 Hex Board. Image from

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Game Theory & Nash Equilibrium

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.

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.

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The Tragic Demise of a Beautiful Mind

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].

Russell Crowe as John Nash in the movie A Beautiful Mind.

Russell Crowe as John Nash in the movie A Beautiful Mind.

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Biscuits, Books, Coins and Cards: Massive Hangovers

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.

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.

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Chess Harmony

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’

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