Python/project_euler/problem_057/sol1.py
Benjamin Smith 58875674da
Project Euler 57 - Square root convergents (#3259)
* include solution for problem 57

* fix line to long errors

* update filenames and code to comply with new regulations

* more descriptive local variables
2020-10-16 21:17:35 +05:30

49 lines
1.5 KiB
Python

"""
Project Euler Problem 57: https://projecteuler.net/problem=57
It is possible to show that the square root of two can be expressed as an infinite
continued fraction.
sqrt(2) = 1 + 1 / (2 + 1 / (2 + 1 / (2 + ...)))
By expanding this for the first four iterations, we get:
1 + 1 / 2 = 3 / 2 = 1.5
1 + 1 / (2 + 1 / 2} = 7 / 5 = 1.4
1 + 1 / (2 + 1 / (2 + 1 / 2)) = 17 / 12 = 1.41666...
1 + 1 / (2 + 1 / (2 + 1 / (2 + 1 / 2))) = 41/ 29 = 1.41379...
The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion,
1393/985, is the first example where the number of digits in the numerator exceeds
the number of digits in the denominator.
In the first one-thousand expansions, how many fractions contain a numerator with
more digits than the denominator?
"""
def solution(n: int = 1000) -> int:
"""
returns number of fractions containing a numerator with more digits than
the denominator in the first n expansions.
>>> solution(14)
2
>>> solution(100)
15
>>> solution(10000)
1508
"""
prev_numerator, prev_denominator = 1, 1
result = []
for i in range(1, n + 1):
numerator = prev_numerator + 2 * prev_denominator
denominator = prev_numerator + prev_denominator
if len(str(numerator)) > len(str(denominator)):
result.append(i)
prev_numerator = numerator
prev_denominator = denominator
return len(result)
if __name__ == "__main__":
print(f"{solution() = }")