problem_125: project euler solution 125

Name: Palindromic sums

The palindromic number 595 is interesting because it can be written as
the sum of consecutive squares: 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2.
There are exactly eleven palindromes below one-thousand that can be written as
consecutive square sums, and the sum of these palindromes is 4164.

Note that 1 = 0^2 + 1^2 has not been included as this problem is concerned with
the squares of positive integers.

Find the sum of all the numbers less than 10^8
that are both palindromic and can be written as the sum of consecutive squares.

Reference: https://projecteuler.net/problem=125

Fixes: #2695

Author: Ravi Kandasamy Sundaram

Diff for: project_euler/problem_125/__init__.py

@@ -0,0 +1,72 @@
+"""
+"https://projecteuler.net/problem=125"
+
+Name: Palindromic sums
+
+
+The palindromic number 595 is interesting because it can be written as
+the sum of consecutive squares: 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2.
+There are exactly eleven palindromes below one-thousand that can be written as
+consecutive square sums, and the sum of these palindromes is 4164.
+
+Note that 1 = 0^2 + 1^2 has not been included as this problem is concerned with
+the squares of positive integers.
+
+Find the sum of all the numbers less than 10^8
+that are both palindromic and can be written as the sum of consecutive squares.
+
+"""
+
+
+import math
+
+
+def gen_square_sums(limit: int) -> list:
+    """
+    Generates and returns thee square sums till n.
+    Using the formula 1^2 + 2^2 + .. + n^2 = n * (n + 1) * (2n + 1) / 6
+    """
+    square_sums = []
+    for n in range(limit):
+        square_sum = (n * (n + 1) * (2 * n + 1)) // 6
+        square_sums.append(square_sum)
+    return square_sums
+
+
+def gen_palindromic_square_sums(square_sums) -> list:
+    """
+    Filters and returns the palindromic square sums
+    Difference between any two index gives the sum of squares in that range.
+    """
+    palindromic_square_sums = set()
+    for i in range(len(square_sums) - 2):
+        for j in range(i + 2, len(square_sums)):
+            diff = square_sums[j] - square_sums[i]
+            if is_palindrome(diff):
+                palindromic_square_sums.add(diff)
+    return sorted(palindromic_square_sums)
+
+
+def is_palindrome(n: int) -> bool:
+    """
+    Returns if the number is palindrome or not
+    """
+    return str(n) == str(n)[::-1]
+
+
+def solution(limit: int = 1e8) -> int:
+    """
+    Returns the sum of palindromic squares sums till 10^8.
+    """
+    square_sums = gen_square_sums(int(math.sqrt(limit)))
+    palindromic_square_sums = gen_palindromic_square_sums(square_sums)
+
+    res = index = 0
+    while palindromic_square_sums[index] <= limit:
+        res += palindromic_square_sums[index]
+        index += 1
+    return res
+
+
+if __name__ == "__main__":
+    print(solution())