## Coins and Change

This is a puzzle.

Imagine the country of Jockland, where coins exist in the following denominations: 3~, 5~, 30~, 50~, 300~, 500~, etc., with ‘~’ their equivalent of the penny. But there is no single-~ coin. If people need to pay 7~, they might offer 2 coins of 5~ and expect to get a 3~ in change.  Can you figure out how to pay a given amount, say 11~, using only these denominations of coins, and getting only these denominations of coins in change?  Can you pay any amount, or are there amounts you couldn’t pay exactly?  Can you think of a quick way to determine how many of each coin it takes to pay a given amount, and how you would get change back?

In the neighboring country of Jerkey, they scoff at the Jocklanders and their system of coins.  Jerkey has coins of 1@, 3@, 9@, 27@, 81@, 100@, 300@, 900@, 2700@, where the ‘@’ is their equivalent of the penny.  How do the people from Jerkey pay for their stuff, e.g. 11@?  Can you pay any amount, or are there amounts you couldn’t pay exactly?  Can you think of a quick way to determine how many of each coin it takes to pay a given amount, and how you would get change back?

In the country of Jammer, storekeepers keep no money in their cash registers, and never give change.  Money paid by the customers goes into a slot and drops directly into a vault.  Store employees in Jammer feel very safe that way, and Jammer is proud of their system.  Jammer’s coins come in denominations of 2&, 5&, 20&, 50&, 200&, 500&, where the ‘&’ is their equivalent of the penny.  How do people in Jammer pay a given amount, like 11&?  Can you pay any amount, or are there amounts you couldn’t pay exactly?  Can you think of a quick way to determine how many of each coin it takes to pay a given amount?