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would not have bothered with the others; why, therefore, should you get anything for them? For Pascal, though, the problem centered around expectation and justice, so his approach was different. He reasons from the money backwards. Let’s say the game is one of even chances, like flipping a coin, and you’ve agreed that the first player to win three games gets the stakes; when the angel appears, you have won two games, your shady opponent one. You could figure the division like this: “There are 64 pistoles on the table. If I had won this next game, they would all be mine; if I had lost, we would be tied and could divide the pot evenly, 32 each; these two likelihoods are equal, so fairness dictates that I split the difference between 64 and 32 and take 48.” The Venetian pockets his 16 with a suppressed oath, but he cannot fault your logic.
    Now if you were instead ahead by 2 games to 0 when the game is interrupted, you could extend this reasoning: “If I had won the next game, I would have gained all 64; if I had lost, we would be at 2 games to 1—which, as I remember, means I should get 48 pistoles . So the fair division is halfway between these possibilities: 56 for me, 8 for him.” Once again, you are being as just as Aristotle; and you are free to go out into the deepening evening, the serene city before you and gold jingling in your purse.
    So far, so good. But can we figure out a general law for interrupted games where you have, say, r points still to make and your opponent has s ? Yes, but to do so we need to take an excursion . . . to the beach, perhaps.
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    Curlews ride the buffeting wind—celestial surfers. The clouds spread in regular ripples like a vast satin quilt. The waves curl, spread, and spring back, as if the ocean were shaking out her hair. Simile reveals pattern—one of the deepest human pleasures, a source of excitement and wonder. For Pascal, the mystical prevalence of pattern was evidence of design and spiritual meaning; for scientists, it is an invitation to explore the unknown, promising that the seemingly random has hidden structure.
    In art, we play with pattern to make our own significant marks. We can start here, scratching the simplest figure, 1, in the sand. On either side, for symmetry, another 1, thus:

    Let’s spread these wings a little wider, using a rule (itself a kind of pattern) for filling the space between them: mark down the sum of the two numbers just above to the left and right:
and so on, filling the sand as we go with our own symmetrical but unexpected design.
    What do we have here? It seems, at first glance, to be no more than the sort of doodle found in the margins of all schoolbooks. Look at it more closely, though—as Pascal did in his Traité du triangle arithmétique —and you will see wonders of pattern. Let’s start by skewing our triangle slightly, to make its rows, columns, and diagonals a little more clear.

    Pattern reveals itself first as a lesson in counting. The left column counts as infants do: “This one and then this one and then this one”; the second as grownups do, adding up as we go. The third column lists what are called triangular numbers , that is, the numbers of dots needed to construct equilateral triangles like these:
(or the number of figures, row by row, in Pascal’s own triangle: 1, 3, 6, 10 . . . see?). The next column lists pyramidal numbers , the number of equal-size stones needed to build a regular pyramid with a triangular base—and so on through Fibonacci series, fractal patterns, and further delights of complexity.
    The second trick of the triangle may have been discovered by the ancient Hindus and Chinese, and was certainly known to Omar Khayyám (an excellent mathematician as well as poet, tentmaker, and philosophical toper). You will probably remember from school how easy it is, when squaring a binomial ( a + b ), to forget to include all the relevant combinations