Recontres, otherwise known as matches, is an old French card game; to call it a game may be something of a stretch of the imagination. It is extremely simple, with the biggest benefit being that it provides an extremely easy way to gamble. This game also is interesting because it is a nice example of how the natural logarithm pops up in several places.
The game itself is quite simple. Lay out the fifty two cards in a standard card deck. Then, take a second deck and lay these, one on top of each card from the first deck. The score is determined by counting the number of matching cards in the two decks.
This game was a very common part of gambling among the commoners in France in the 16th through the 18th centuries. Bets were placed on whether or not there would be an exact match in the fifty two pairs, or how many matches would have the same suit or the same face value.
Eventually, as this game reached its popular peak around 1700, mathematicians became interested in the problems that this game presented. In 1708, the French mathematician Pierre Raymond de Montmort posed le probleme de recontres, which broke the game down to a simple question: what is the probability that no matches take place in a game of recontres?
Montmort went on to solve his own problem by incorporating the idea of derangement into a solution. A derangement is a permutation of objects that leaves no object in its original location. A derangement is mathematically defined as follows, with n being the number of objects that are deranged:
1 1 1 (-1)^n
D(n) = n! * (1 - ---- + ---- - ---- + ... + ------ )
1! 2! 3! n!
Now, recontres is essentially a real-world example of a derangement of size 52. You can quickly check the actual derangement by seeing how many of the pairs match up. The actual numerical value of D(52) is then the number of possible derangements in a set of size 52, and 52! is the number of total arrangements. The odds that a derangement will occur in the game of recontres is then D(52)/52!.
Now, if we whip out our handy calculator and calculate D(52)/52!, we will get a number roughly equal to 0.3679. This means that no matches will occur in a game of recontres 36.79% of the time.
But it gets more mathematically interesting, and it involves the number e, otherwise known as the natural logarithm. You see 1/e to four decimal places is equal to 0.3679. Montmort immediately realized this and applied it to a new definition of e:
1 D(infinity)
- = -----------
e infinity!
In fact, a sufficiently large number to replace infinity above leads to a very good approximation of e.
Recontres is a simple card game that also provides a clear lesson in how mathematics can reduce card games to number crunching, and also provides a great example of the natural logarithm popping up in the real world.