Solution for the Euler Project Problem 122

project_euler/problem_122/sol1.py

@@ -0,0 +1,65 @@
+"""
+Project Euler Problem 122: https://projecteuler.net/problem=122
+
+Efficient Exponentiation
+
+It uses the fact that for rather small n, applicable for this problem, the solution
+for each number
+can be formed by increasing the largest element.
+
+References:
+- https://en.wikipedia.org/wiki/Addition_chain
+
+>>> solution(14)
+45
+"""
+
+
+def solve(nums: list[int], goal: int, depth: int) -> bool:
+    """
+    Checks if nums can can have a sum equal to goal, given that length of nums does
+    not exceed depth.
+
+    >>> solve([1], 2, 2)
+    True
+    >>> solve([1], 2, 0)
+    False
+    """
+    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:
+    """
+    Calculates sum of smallest number of multiplactions for each number up to
+    and including n.
+
+    >>> solution(1)
+    0
+    >>> solution(2)
+    1
+    >>> solution(15)
+    50
+    """
+    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() = }")