You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth 1, 2, 5, 10, and 20 florins, but you have no idea which coin is which and you don’t speak the local language. You find a vending machine where a single candy can be bought for 1 florin: you insert any kind of coin, and receive
1 candy plus any change owed. You can only buy one candy at a time, but you can buy as many as you want, one after the other.
What is the least number of candies that you must buy to ensure that you can determine the values of all the coins? Prove that your answer is correct.