mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-30 16:31:08 +00:00
93fb555e0a
* Enable ruff SIM102 rule * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Fix * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
101 lines
2.8 KiB
Python
101 lines
2.8 KiB
Python
"""
|
|
|
|
Project Euler Problem 207: https://projecteuler.net/problem=207
|
|
|
|
Problem Statement:
|
|
For some positive integers k, there exists an integer partition of the form
|
|
4**t = 2**t + k, where 4**t, 2**t, and k are all positive integers and t is a real
|
|
number. The first two such partitions are 4**1 = 2**1 + 2 and
|
|
4**1.5849625... = 2**1.5849625... + 6.
|
|
Partitions where t is also an integer are called perfect.
|
|
For any m ≥ 1 let P(m) be the proportion of such partitions that are perfect with
|
|
k ≤ m.
|
|
Thus P(6) = 1/2.
|
|
In the following table are listed some values of P(m)
|
|
|
|
P(5) = 1/1
|
|
P(10) = 1/2
|
|
P(15) = 2/3
|
|
P(20) = 1/2
|
|
P(25) = 1/2
|
|
P(30) = 2/5
|
|
...
|
|
P(180) = 1/4
|
|
P(185) = 3/13
|
|
|
|
Find the smallest m for which P(m) < 1/12345
|
|
|
|
Solution:
|
|
Equation 4**t = 2**t + k solved for t gives:
|
|
t = log2(sqrt(4*k+1)/2 + 1/2)
|
|
For t to be real valued, sqrt(4*k+1) must be an integer which is implemented in
|
|
function check_t_real(k). For a perfect partition t must be an integer.
|
|
To speed up significantly the search for partitions, instead of incrementing k by one
|
|
per iteration, the next valid k is found by k = (i**2 - 1) / 4 with an integer i and
|
|
k has to be a positive integer. If this is the case a partition is found. The partition
|
|
is perfect if t os an integer. The integer i is increased with increment 1 until the
|
|
proportion perfect partitions / total partitions drops under the given value.
|
|
|
|
"""
|
|
|
|
import math
|
|
|
|
|
|
def check_partition_perfect(positive_integer: int) -> bool:
|
|
"""
|
|
|
|
Check if t = f(positive_integer) = log2(sqrt(4*positive_integer+1)/2 + 1/2) is a
|
|
real number.
|
|
|
|
>>> check_partition_perfect(2)
|
|
True
|
|
|
|
>>> check_partition_perfect(6)
|
|
False
|
|
|
|
"""
|
|
|
|
exponent = math.log2(math.sqrt(4 * positive_integer + 1) / 2 + 1 / 2)
|
|
|
|
return exponent == int(exponent)
|
|
|
|
|
|
def solution(max_proportion: float = 1 / 12345) -> int:
|
|
"""
|
|
Find m for which the proportion of perfect partitions to total partitions is lower
|
|
than max_proportion
|
|
|
|
>>> solution(1) > 5
|
|
True
|
|
|
|
>>> solution(1/2) > 10
|
|
True
|
|
|
|
>>> solution(3 / 13) > 185
|
|
True
|
|
|
|
"""
|
|
|
|
total_partitions = 0
|
|
perfect_partitions = 0
|
|
|
|
integer = 3
|
|
while True:
|
|
partition_candidate = (integer**2 - 1) / 4
|
|
# if candidate is an integer, then there is a partition for k
|
|
if partition_candidate == int(partition_candidate):
|
|
partition_candidate = int(partition_candidate)
|
|
total_partitions += 1
|
|
if check_partition_perfect(partition_candidate):
|
|
perfect_partitions += 1
|
|
if (
|
|
perfect_partitions > 0
|
|
and perfect_partitions / total_partitions < max_proportion
|
|
):
|
|
return int(partition_candidate)
|
|
integer += 1
|
|
|
|
|
|
if __name__ == "__main__":
|
|
print(f"{solution() = }")
|