 However, Pascal was able to use it as a starting point for developing more advanced computational methods.Through clever manipulation of identities involving what is today known as Pascal's triangle (including several of the first explicit proofs by induction) Pascal finally showed that in a game where one player needs "r" points to win and the other needs "s" points to win, the correct division of the stakes is in the ratio of (using modern notation)

The problem of dividing the stakes became a major motivating example for Pascal in his "Treatise on the arithmetic triangle".Though Pascal's derivation of this result was independent of Fermat's tabular method, it is clear that it also describes exactly the counting of different outcomes of formula_1 additional rounds that Fermat suggested.