Optimize making_change: add cache and math check for better performance

Author: AFatmaa

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -1,22 +1,38 @@
 from typing import List
 
+cache = {}
 
 def ways_to_make_change(total: int) -> int:
     """
     Given access to coins with the values 1, 2, 5, 10, 20, 50, 100, 200, returns a count of all of the ways to make the passed total value.
 
     For instance, there are two ways to make a value of 3: with 3x 1 coins, or with 1x 1 coin and 1x 2 coin.
     """
+    # Clear cache before starting new calculation
+    cache.clear()
     return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1])
 
 
 def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
     """
     Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
     """
+    # We use tuple(coins) because lists cannot be keys.
+    key = (total, tuple(coins))
+
+    if key in cache:
+        return cache[key]
+    
     if total == 0 or len(coins) == 0:
         return 0
 
+    # Check if only one coin type remains
+    if len(coins) == 1:
+        if total % coins[0] == 0:
+            return 1
+        else:
+            return 0
+        
     ways = 0
     for coin_index in range(len(coins)):
         coin = coins[coin_index]
@@ -29,4 +45,7 @@ def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
                 intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:])
                 ways += intermediate
             count_of_coin += 1
+            
+    # Store result in cache
+    cache[key] = ways
     return ways