from fractions import Fraction


def euler_100():
    F_ONE_HALF = Fraction(1, 2)
    prev_blue_discs, blue_discs = 10, 10
    while True:
        # Brute force search blue discs
        blue_discs += 1

        # Binary search red discs
        red_upper = blue_discs // 2
        red_lower = 1
        while red_upper != red_lower:
            red_discs = red_lower + (red_upper - red_lower) // 2
            total_discs = blue_discs + red_discs
            f1 = Fraction(blue_discs, total_discs)
            f2 = Fraction(blue_discs - 1, total_discs - 1)
            f = f1 * f2
            if f == F_ONE_HALF:
                # End criterion by Euler Problem
                if total_discs > 10**12:
                    return blue_discs

                # The ration between two solutions seems to be constant so we
                # can get the next larger candidate for blue_discs much
                # quicker.
                ratio = blue_discs / prev_blue_discs
                # print("blue_discs={} ratio={}".format(blue_discs, ratio))
                prev_blue_discs = blue_discs
                blue_discs = int(blue_discs * ratio)
                break
            elif f < F_ONE_HALF:
                red_upper = red_discs
            else:
                red_lower = red_discs

            # No solution for current number of blue discs
            if abs(red_upper - red_lower) <= 1:
                break


if __name__ == "__main__":
    solution = euler_100()
    print("e100.py: " + str(solution))
    assert(solution == 756872327473)