# Pierre de Fermat

### b. Beaumont-de-Lomagne, France 20 August 1601 d. Castres, France 12 January 1665

Fermat received a degree of Bachelor of Civil Laws from the University of Orleans on 1 May 1631. He purchased the offices of conseiller and commissaire aux requêtes in the parlement of Toulouse. It was here he became friends with Carcavi. Over the years Fermat rose to the highest councils in the parlement.

At some time in 1654 Pascal apparently sent a letter to Fermat in which he asked that Fermat confirm his reasoning concerning one or more problems in gaming. A sequence of letters between the two men followed in which the principle element of interest is the solution to the problem of points. In the course of these letters, three methods of solution were proposed. That of Fermat, a direct method, is methodologically quite simple, but difficult to apply in all but the easiest cases. Those of Pascal are of much greater generality and more subtle. The content of the letters is outlined below.

Letter 1. Fermat replies to Pascal, undated.

A wager is made between two players such that if the first should cast at least one six in eight rolls of a fair die he would win the amount wagered. Suppose the first has rolled three times and not yet made his point. The opponent proposes that the player not roll the fourth time. What is the value of this forsaken roll?

Pascal had claimed that the first player should receive of the stake based upon the erroneous assumption that the player had agreed to give up the previous three throws as well. But this is not the same problem as that posed. Fermat observes that since the entire stake is still in play the player should receive of the stake just as if he had agreed to forsake the first throw.

Letter 2. Pascal to Fermat, 29 July 1654.

Pascal wrote again to Fermat on Wednesday, 29 July 1654. He admitted his error with the dice problem and then took up discussion of the problem of points. They had been communicating previously concerning this problem and each had found a solution.

Suppose players A and B each stake the same amount. They agree to play until one of them has won n games. The winner receives the entire amount wagered. However, the game is terminated when Player A has won a game and Player B has won b. How should the stakes be divided?

Pascal first described a recursive procedure which allowed him to work backwards from the game in which Player A has n-1 games and Player B has n-2 to the current state where Player A has a points and Player B has b. Note that when Player A has n-1 games and Player B has n-2, the game either terminates in the next round or the players tie thereby making the division of stakes easy: if A wins, he takes all, if he loses, the stake is divided, so A should receive 3/4 of the stake.

Pascal then took up the more difficult task of directly computing the value of the game when Player A has won 1 game and Player B none. Here reference is made to theorems contained in his Treatise on the Arithmetic Triangle.

Lastly, Pascal remarked on de Méré's problem of the dice.

The reply of Fermat to this letter is not extant.

Letter 3. Pascal to Fermat, 24 August 1654.

There is evidence that Fermat wrote again to Pascal but this letter does not exist.

Letter 4. Pascal to Fermat, Monday, 24 August 1654.

In this letter Pascal objected to Fermat's method of solution.

Fermat began with the current state of the game. Since Player A lacks n-a games and Player B lacks n-b, the game will be decided in 2n-a-b-1 rounds. He examined all possible sequences of rounds of length enumerating those sequences which constitute a win for Player A and those which constitute a win for Player B.

Pascal noted that the actual game may terminate in fewer rounds than 2n-a-b-1. Therefore these sequences do not represent all possible futures because some sequences would never be realized. Pascal reported that Roberval objected to this method on the basis of these unrealizable sequences. Pascal objected too that this method does not generalize to 3 or more players. But this was due to his misinterpretation of Fermat's method.

Pascal presented an example in which certain sequences produce two winners rather than one because he ignored the fact that complete sequences need not be realized. The game is over when someone wins. Due to his failure to handle these cases correctly, Fermat's method is made to appear to yield an incorrect solution.

Letter 5. Fermat to Pascal, 25 September 1654.

In a letter dated 29 August 1654, Fermat confirmed Pascal's solution to the example with 3 players discussed in the previous letter. No reply was made by Pascal.

Letter 6. Fermat to Pascal, 25 September 1654.

Fermat remarked at length on the correctness of his method. He pointed out that one must take into consideration the order in which wins occur. Moreover, there is no harm in considering sequences which are not realizable. They exist solely to make the enumeration of cases easy.

Here is a link to my translation of the correspondence