Class for sorting pool balls
I think it is usually better to take advantage of the built-in functions. Python lists already have a sort function, so you would expect a list of pool balls to be sortable by the start arrangement (a list of tuples/lists is already sortable, Python compares them element wise).

One way to achieve that is with a custom key function:

order = [1, 13, 11, 7, 8, 6, 9, 5, 14, 15, 2, 12, 3, 10, 4]
# Simplified balls list, as the fill style is not really needed for this
balls = [[n] for n in range(1, 16)]
balls.sort(key=lambda ball: order.index(ball[0]))


This is very succinct, but you could also promote the lambda key function to a proper function:

def pool_start_order(ball):
    order = [1, 13, 11, 7, 8, 6, 9, 5, 14, 15, 2, 12, 3, 10, 4]
    return order.index(ball[0])

balls = [[n] for n in range(1, 16)]
balls.sort(key=pool_start_order)


Another alternative is to make a PoolBall class, that is sortable, something like this:

class PoolBall(object):
    order = [1, 13, 11, 7, 8, 6, 9, 5, 14, 15, 2, 12, 3, 10, 4]

    def __init__(self, number):
        assert 1 &lt;= number &lt;= 15
        self.number = number
        self.fill = 'Solid' if number &lt;= 8 else 'Striped'
        self.order = PoolBall.order.index(number)

    def __lt__(self, other):
        return self.order &lt; other.order

    def __eq__(self, other):
        return self.number == other.number

    def __str__(self):
        return "{self.number} {self.fill}".format(self=self)

    def __repr__(self):
        return "{self.__class__.__name__}({self.number})".format(self=self)


    balls = [PoolBall(i) for i in range(1, 16)]
    balls.sort()


Here, the __str__ and __repr__ are just there for pretty printing.

Whichever way you go, once you have a correctly sorted list of balls, you can use a function that successively takes one more ball, something like:

def pyramid_arrangement(balls):
    """
    Given a list of balls, arranges them in a pyramid shape.
    Yields each row.

    Leaves the balls list empty.
    """
    take = 1
    while balls:
        yield [balls.pop() for _ in range(take)]
        take += 1


Which you can use like this:

if __name__ == "__main__":
    balls = [PoolBall(i) for i in range(1, 16)]
    balls.sort(reverse=True)
    for row in pyramid_arrangement(balls):
        print row