Add initial version for euler project problem 122.

Author: mindaugl

Diff for: project_euler/problem_122/__init__.py

@@ -0,0 +1,35 @@
+"""
+Project Euler Problem 122: https://projecteuler.net/problem=122
+
+Efficient Exponentiation
+"""
+
+
+def solve(nums: list[int], goal: int, depth: int) -> bool:
+    if len(nums) > depth:
+        return False
+    for el in nums:
+        if el + nums[-1] == goal:
+            return True
+        nums.append(el + nums[-1])
+        if solve(nums, goal, depth):
+            return True
+        del nums[-1]
+    return False
+
+
+def solution(n: int = 200) -> int:
+    m = [0]
+    for i in range(2, n + 1):
+        max_length = 0
+        while True:
+            nums = [1]
+            max_length += 1
+            if solve(nums, i, max_length):
+                break
+        m.append(len(nums))
+    return sum(m)
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")