diff --git a/project_euler/problem_057/__init__.py b/project_euler/problem_057/__init__.py new file mode 100644 index 000000000..e69de29bb diff --git a/project_euler/problem_057/sol1.py b/project_euler/problem_057/sol1.py new file mode 100644 index 000000000..04b6199f4 --- /dev/null +++ b/project_euler/problem_057/sol1.py @@ -0,0 +1,48 @@ +""" +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() = }")