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?*

We assume that each square has sides of unit length. A coin of radius *r* just touches the edge of the unit square if its centre lies on an inner square of side-length (1 – 2 *r*). To win, the coin centre must land within the smaller square. Clearly, the smaller *r* is, the greater the chance of winning.

We assume that all points within the unit square are equally likely, so the probability of the centre being within the inner square is ( 1 – 2 *r *) ^{2 }. For a 50% chance of victory, we have

( 1 – 2 *r *) ^{2 } = ½ or 4 r ^{2} – 4 r + ½ = 0

This quadratic equation has two roots, one greater than ½ and one less than ½. Only the latter is of interest:

r = ( 2 – √ 2 ) / 4 ≈ 0.146 ≈ 1 / 7 .

So, if the coin radius is about one-seventh the side-length of the square, there is a roughly 50% chance of winning. In many fair-grounds, coins or tokens are markedly larger than this!

**Buffon and his Needle**

When Georges-Louis Leclerc, Compte de Buffon (1707 – 1788) was just 20, he discovered the binomial theorem (for integer exponents) independently of Newton. He was greatly interested in probability, approaching that topic – like many other scholars – from its connections with gambling. He considered the question: can one be sure of winning in roulette by repeatedly doubling one’s wager until one is in profit? This is related to the notorious ‘Gambler’s Ruin‘. Buffon’s 1733 *M**é**moire sur le jeu de franc-carreau* earned him election to the Royal Academy of Sciences. He published extensively, including 36 volumes of an encyclopedia on the natural sciences.

Laplace discussed Buffon’s work in his *Théorie analytique des probabilités *(1812). Buffon was the first mathematician to deal with random trials. This was in his work on franc-carreau and similar games, which was a clever combination of ideas from two distinct areas, geometry and probability. It can be considered the origin of geometric probability. His study of franc-carreau led immediately to the problem now called Buffon’s Needle [See article on Buffon’s Needle on this blog]

For an interesting account of Buffon’s life and work, see the article *Georges-Louis Leclerc Buffon* (1707–1788), by Heinz Klaus Strick, on the Mathematics-in-Europe web site.

**Sources **

Heinz Klaus Strick: *Georges-Louis Leclerc Buffon *at Mathematics-in-Europe

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