From c585cb122718e219870ea7a2af110939b55e52f9 Mon Sep 17 00:00:00 2001
From: Mindaugas <76015221+mindaugl@users.noreply.github.com>
Date: Fri, 18 Apr 2025 07:16:15 +0800
Subject: [PATCH] Solution for the Euler Project problem 136 (#12658)

* Add initial version of file for the Euler project problem 136 solution.

* Add documentation and tests for the Euler project problem 136 solution.

* Update sol1.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update sol1.py

* Update sol1.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update sol1.py

* Update sol1.py

* Update sol1.py

* Update sol1.py

* Update sol1.py

* Update sol1.py

---------

Co-authored-by: Maxim Smolskiy <mithridatus@mail.ru>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
---
 project_euler/problem_136/__init__.py |  0
 project_euler/problem_136/sol1.py     | 63 +++++++++++++++++++++++++++
 2 files changed, 63 insertions(+)
 create mode 100644 project_euler/problem_136/__init__.py
 create mode 100644 project_euler/problem_136/sol1.py

diff --git a/project_euler/problem_136/__init__.py b/project_euler/problem_136/__init__.py
new file mode 100644
index 000000000..e69de29bb
diff --git a/project_euler/problem_136/sol1.py b/project_euler/problem_136/sol1.py
new file mode 100644
index 000000000..688a9a5d7
--- /dev/null
+++ b/project_euler/problem_136/sol1.py
@@ -0,0 +1,63 @@
+"""
+Project Euler Problem 136: https://projecteuler.net/problem=136
+
+Singleton Difference
+
+The positive integers, x, y, and z, are consecutive terms of an arithmetic progression.
+Given that n is a positive integer, the equation, x^2 - y^2 - z^2 = n,
+has exactly one solution when n = 20:
+                              13^2 - 10^2 - 7^2 = 20.
+
+In fact there are twenty-five values of n below one hundred for which
+the equation has a unique solution.
+
+How many values of n less than fifty million have exactly one solution?
+
+By change of variables
+
+x = y + delta
+z = y - delta
+
+The expression can be rewritten:
+
+x^2 - y^2 - z^2 = y * (4 * delta - y) = n
+
+The algorithm loops over delta and y, which is restricted in upper and lower limits,
+to count how many solutions each n has.
+In the end it is counted how many n's have one solution.
+"""
+
+
+def solution(n_limit: int = 50 * 10**6) -> int:
+    """
+    Define n count list and loop over delta, y to get the counts, then check
+    which n has count == 1.
+
+    >>> solution(3)
+    0
+    >>> solution(10)
+    3
+    >>> solution(100)
+    25
+    >>> solution(110)
+    27
+    """
+    n_sol = [0] * n_limit
+
+    for delta in range(1, (n_limit + 1) // 4 + 1):
+        for y in range(4 * delta - 1, delta, -1):
+            n = y * (4 * delta - y)
+            if n >= n_limit:
+                break
+            n_sol[n] += 1
+
+    ans = 0
+    for i in range(n_limit):
+        if n_sol[i] == 1:
+            ans += 1
+
+    return ans
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")