Problem Statement
In Byteland they have a very strange monetary system.
Each Bytelandian gold coin has an integer number written on it. A coin n can be exchanged in a bank into three coins: n/2, n/3 and n/4. But these numbers are all rounded down (the banks have to make a profit).
You can also sell Bytelandian coins for American dollars. The exchange rate is 1:1. But you can not buy Bytelandian coins.
You have one gold coin. What is the maximum amount of American dollars you can get for it?
Input
The input will contain several test cases (not more than 10). Each test case is a single line with a number n, 0 <= n <= 1 000 000 000. It is the number written on your coin.
Output
For each test case output a single line, containing the maximum amount of American dollars you can make.
Example
Input:
12
2
Output:
13
2
You can change 12 into 6, 4 and 3, and then change these into 6+6+4+3=3=13. If you try changing the coin 2 into 3 smaller coins, you will get 1, 0 and 0, and later you can get no more than 1 coin so it is better to exchange 2 coins for 2 dollars.
Approach to Solution
This problem is a classic example of Recursive Programming. Here the main idea is to get the maximum sum of coins. For cases where N > number of coins in (N/2+N/3+N/3) we will simply get the change for the exact coin N.
# Dictionary to store the number of coins to their corresponding keys
array = {0:0,1:1}
def solve(n):
if n in array:
return array[n] #retruns the value of the key if found in array
else:
array[n] = max(n,solve(n//2)+solve(n//3)+solve(n//4))
return array[n]
if __name__ == '__main__':
# taking the input
i = 10
while(i > 0):
n = int(input())
print(solve(n))
i -= 1
Problem link – link
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