i tried running my change_fast again change (from the recitation) for comparison of efficiency.
In some cases, i saw that my change_fast is working slower. So, i tired to print my dictionary every call to see if there's a problem in my implementation of memoization. i saw that not only the dictionary was not consistently growing at each print, but it also didn't present in recursive call X, some of the keys that were in the dictionary in call X-1.
1. the keys that were presented at each call were the ones required for the calculation of the result of the "closest" waiting call.
2. the dictionary wasn't in the same size at each print.
does this mean i didn't implement the memoization properly?