Algorithm to determine coin combinations

Question

I was recently faced with a prompt for a programming algorithm that I had no idea what to do for. I\'ve never really written an algorithm before, so I\'m kind of a newb at t

Answer 1

This seems somewhat like a Partition, except that you don't use all integers in 1:50. It also seems similar to bin packing problem with slight differences:

• Wikipedia: Partition (Number Theory)
• Wikipedia: Bin packing problem
• Wolfram Mathworld: Partiton

Actually, after thinking about it, it's an ILP, and thus NP-hard.

I'd suggest some dynamic programming appyroach. Basically, you'd define a value "remainder" and set it to whatever your goal was (say, 50). Then, at every step, you'd do the following:

1. Figure out what the largest coin that can fit within the remainder
2. Consider what would happen if you (A) included that coin or (B) did not include that coin.
3. For each scenario, recurse.

So if remainder was 50 and the largest coins were worth 25 and 10, you'd branch into two scenarios:

1. Remainder = 25, Coinset = 1x25
2. Remainder = 50, Coinset = 0x25

The next step (for each branch) might look like:

1-1. Remainder = 0,  Coinset = 2x25 <-- Note: Remainder=0 => Logged
1-2. Remainder = 25, Coinset = 1x25
2-1. Remainder = 40, Coinset = 0x25, 1x10
2-2. Remainder = 50, Coinset = 0x25, 0x10

Each branch would split into two branches unless:

• the remainder was 0 (in which case you would log it)
• the remainder was less than the smallest coin (in which case you would discard it)
• there were no more coins left (in which case you would discard it since remainder != 0)