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* add solution to Project Euler problem 206 * Add solution to Project Euler problem 205 * updating DIRECTORY.md * updating DIRECTORY.md * Revert "Add solution to Project Euler problem 205" This reverts commit64e3d36cab. * Revert "add solution to Project Euler problem 206" This reverts commit53568cf4ef. * add solution for project euler problem 207 * updating DIRECTORY.md * add type hint for output of helper function * Correct default parameter value in solution * use descriptive variable names and remove problem solution from doctest Fixes: #2695 Co-authored-by: nico <esistegal-aber@gmx.de> Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
99 lines
2.7 KiB
Python
99 lines
2.7 KiB
Python
"""
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Project Euler Problem 207: https://projecteuler.net/problem=207
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Problem Statement:
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For some positive integers k, there exists an integer partition of the form
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4**t = 2**t + k, where 4**t, 2**t, and k are all positive integers and t is a real
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number. The first two such partitions are 4**1 = 2**1 + 2 and
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4**1.5849625... = 2**1.5849625... + 6.
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Partitions where t is also an integer are called perfect.
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For any m ≥ 1 let P(m) be the proportion of such partitions that are perfect with
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k ≤ m.
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Thus P(6) = 1/2.
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In the following table are listed some values of P(m)
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P(5) = 1/1
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P(10) = 1/2
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P(15) = 2/3
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P(20) = 1/2
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P(25) = 1/2
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P(30) = 2/5
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...
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P(180) = 1/4
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P(185) = 3/13
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Find the smallest m for which P(m) < 1/12345
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Solution:
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Equation 4**t = 2**t + k solved for t gives:
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t = log2(sqrt(4*k+1)/2 + 1/2)
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For t to be real valued, sqrt(4*k+1) must be an integer which is implemented in
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function check_t_real(k). For a perfect partition t must be an integer.
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To speed up significantly the search for partitions, instead of incrementing k by one
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per iteration, the next valid k is found by k = (i**2 - 1) / 4 with an integer i and
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k has to be a positive integer. If this is the case a partition is found. The partition
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is perfect if t os an integer. The integer i is increased with increment 1 until the
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proportion perfect partitions / total partitions drops under the given value.
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"""
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import math
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def check_partition_perfect(positive_integer: int) -> bool:
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"""
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Check if t = f(positive_integer) = log2(sqrt(4*positive_integer+1)/2 + 1/2) is a
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real number.
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>>> check_partition_perfect(2)
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True
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>>> check_partition_perfect(6)
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False
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"""
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exponent = math.log2(math.sqrt(4 * positive_integer + 1) / 2 + 1 / 2)
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return exponent == int(exponent)
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def solution(max_proportion: float = 1 / 12345) -> int:
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"""
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Find m for which the proportion of perfect partitions to total partitions is lower
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than max_proportion
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>>> solution(1) > 5
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True
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>>> solution(1/2) > 10
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True
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>>> solution(3 / 13) > 185
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True
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"""
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total_partitions = 0
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perfect_partitions = 0
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integer = 3
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while True:
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partition_candidate = (integer ** 2 - 1) / 4
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# if candidate is an integer, then there is a partition for k
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if partition_candidate == int(partition_candidate):
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partition_candidate = int(partition_candidate)
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total_partitions += 1
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if check_partition_perfect(partition_candidate):
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perfect_partitions += 1
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if perfect_partitions > 0:
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if perfect_partitions / total_partitions < max_proportion:
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return partition_candidate
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integer += 1
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if __name__ == "__main__":
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print(f"{solution() = }")
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