Add first solution for Project Euler Problem 207 (#3522)

* 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 commit 64e3d36cab.

* Revert "add solution to Project Euler problem 206"

This reverts commit 53568cf4ef.

* 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>

DIRECTORY.md

@@ -720,6 +720,8 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_174/sol1.py)
   * Problem 191
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_191/sol1.py)
+  * Problem 207
+    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_207/sol1.py)
   * Problem 234
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_234/sol1.py)
   * Problem 551

project_euler/problem_207/sol1.py

@@ -0,0 +1,98 @@
+"""
+
+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:
+            if perfect_partitions / total_partitions < max_proportion:
+                return partition_candidate
+        integer += 1
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")