How To Design A Franc-carreau or Fair-square Game

The Franc-Carreau or fair-square is a game of chance from a time when the mathematical notion of probability was just seeping into the world of gambling. Back in the 18th Century, Georges-Louis LeClerc, Comte de Buffon of Burgundy, France, was aiming to make it into the Royal Academy of Sciences in Paris as a scholar-member.

He had completed a law degree, but was too fascinated by science to start his career in law. After encountering Swiss mathematician Gabriel Cramer, Buffon decided that pure mathematics was going to be his vehicle forward.

In order to follow his dream, Buffon presented a novel paper that combined the notions of geometry and probability. Scholars of the time had widely considered these two fields unrelated to each other. In his paper, Buffon started with designing a humble game of Franc-Carreau. How exactly did this game juxtapose geometry and probability? Let us find out.

This essay is supported by Generatebg

The Franc-Carreau or Fair-square Game

The term “Franc-Carreau” contextually translates to “tile within a tile” or “square within a square”. The Franc-Carreau game involved a coin (back then, the ecu was used for this purpose) and a square grid-like board or tiles.

When a player tosses the coin onto the grid, if it lands completely within a square, the player wins. If the coin happens to touch one or more of the grid-lines, then the player loses.

When you think about it, it is quite a simple game of chance actually. Below, you can see an illustration where the pink coin has landed completely within a square, whereas the green coin touches more than one of the grid lines.

Given this setting, Buffon wanted to figure out the answer to the following question:

What should be the dimensions of the grid/coin such that the Franc-Carreau game is fair, that is, it offers a 50% chance of winning?

---

Buffon’s Approach to the Franc-Carreau or Fair-square Game

To figure out the dimensions of the grid/coin for a fair-square game, Buffon considered a smaller square inside a bigger square such that it was equally spaced from each edge of the bigger square. Let us say that each square in the grid has a length ‘L’.

If this is the case, then a circular coin of radius ‘r’ would just be touching one of the edges of the grid if its centre lands on one of the edges of a smaller square of length (L-2r). The reasoning behind this assertion is visually illustrated below:

As you can see, each edge of the smaller square is at a distance of ‘r’ from the corresponding edge of the bigger square. So, each edge of the smaller square is of length (L-2r).

Whenever the coin’s centre lands inside this smaller square, the player would win. Conversely, whenever the coin’s centre lands outside this smaller square, it would touch at least one of the grid lines and the player would lose.

Having established this setup, Buffon then introduced the notion of probability into the geometric puzzle.

How Buffon Designed the Franc-Carreau or Fair-Square Game

Buffon wanted to compute the probability of winning the Franc-Carreau game using the geometric notion of areas. Let us say that you wish to bet that the coin is to squarely land inside the smaller square.

The probability of this happening is just the ratio of the area of the smaller square divided by the area of the bigger square:

Probability of the coin squarely landing inside the smaller square = (L-2r)²/L²

To design a Fair-square game, all we need to do is assert that this probability equals ½, which results in the following equations:

(L-2r)²/L² = ½

This leads to a quadratic equation with two roots. Considering the relevant root, we arrive at the following solution:

L = (4 + 2√2)r

Thus, Buffon showed that as long as this linear relationship is maintained, the Franc-Carreau or Fair-square game would offer the player a fair 50% chance of winning.

Final Remarks

Although this happens to be a simple game of chance, Buffon’s paper is widely considered as the origin of geometric probability. Furthermore, this was just an appetizer as far as Buffon was concerned.

Using the Franc-Carreau game as the basis, he went on to solve much more advanced geometric probability puzzles in his paper, some of which I plan to cover in a future essay.

---

Reference and credit: Jordon Ellenberg.

If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you: How To Benefit From Braess’s Paradox? and How To Really Benefit From Curves Of Constant Width?

If you would like to support me as an author, consider contributing on Patreon.