Add doctests and documentation for the project euler problem 122.

Author: mindaugl

Diff for: project_euler/problem_122/sol1.py

@@ -2,10 +2,29 @@
 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:
@@ -19,6 +38,17 @@ def solve(nums: list[int], goal: int, depth: int) -> bool:
 
 
 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