B. Bytelandian Gold Coins
time limit per test: 500 ms
memory limit per test: 256 mB
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+$4+$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 out of them. It is better just to change the 2 coin directly into $2.
Solution
Try to solve it on your own before looking at the solution.
Python3 / Python2 / PyPy / ...
# O(ammount of unique states).
import sys
# Dictionary to store already computed results
memo = {0: 0}
def max_dollars(n):
# If it has already been computed, return it directly
if n in memo:
return memo[n]
# Best option:
# keep the coin or exchange it
memo[n] = max(
n,
max_dollars(n // 2) +
max_dollars(n // 3) +
max_dollars(n // 4)
)
return memo[n]
# Read all test cases until EOF
for line in sys.stdin:
n = int(line.strip())
print(max_dollars(n))You are welcome to share your solution in another programming language