"""
Counting Fractions in a range
Project Euler Problem #73
by Muaz Siddiqui

Consider the fraction, n/d, where n and d are positive integers. If n<d and 
HCF(n,d)=1, it is called a reduced proper fraction.

If we list the set of reduced proper fractions for d ≤ 8 in ascending order of 
size, we get:

1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 
3/4, 4/5, 5/6, 6/7, 7/8

It can be seen that there are 3 fractions between 1/3 and 1/2.

How many fractions lie between 1/3 and 1/2 in the sorted set of reduced proper 
fractions for d ≤ 12,000?
"""
from euler_helpers import timeit, totient

@timeit
def answer():
    # The farey sequence has the property for the index that :
    # I_n(h/k) = len(F_n) - 1 - I_n(k-h/k)
    # So count from 1/2 up to 2/3
    count = 0
    a, b, c, d = 1, 2, 6000, 11999
    while not (c == 2 and d == 3):
        count += 1
        k = int((12000 + b)/d)
        a, b, c, d = c, d, k*c - a, k*d - b  
    return count