Communication: magician example

Imagine that you and I are both magicians, performing a trick where I think of a card from a deck of cards and then leave the room, at which point you enter the room and have to guess which card I picked. To perform this trick, I leave some Information in the room that allows you to figure out which card I picked. I do this by encoding a message into the environment (using objects that can carry data). Let’s say that our props table includes (among other things) a coin and two dice (one red, one blue). Assume I have the ability to (without the audience noticing) place the coin and the two dice however I like before leaving the room. Those three objects have a Data capacity of \(\log_2(2 \times 6 \times 6) \approx 6.17\) bits. I can use those objects to represent the encoding of the message I want to send to you.

The messages that I want to send are messages saying which card I’m thinking of. Example messages include the message saying that I’m thinking of the ace of spades, or messages saying that I’m thinking of the king of hearts.

The usual way that humans encode messages like these is to use physical objects like pressure waves in air or pixels on a screen or ink on paper arranged to look like the following patterns: “my card is the ace of spades” or “my card is the king of hearts.” (Don’t confuse the message with the encoding.)

Those sentences are pretty long, though: 28 and 29 letters, respectively. A coin and two dice can’t carry 29 characters. Fortunately, we don’t need all those characters: I can easily encode a message about which card I’m thinking of using only two characters, a rank and a suit, such as \(A♠\) or \(K♡.\) and so on. However, the coin and dice don’t naturally carry rank and suit symbols on them.

However, there are only 52 possible messages I might want to encode, and the coin and dice can be put into \(2 \cdot 6 \cdot 6 = 72\) different configurations. Thus, if we’re clever, and choose a decoding scheme ahead of time, then I can set those objects in such a way that when you see them, you know which card I’m thinking of. Let’s say we’ve agreed on the following decoding scheme:

If the coin is heads, the card is red. If the coin is tails, the card is black.
If the red die is a 1, 2, or 3 then the suit is either spades or hearts. Otherwise, the suit is either clubs or diamonds.
If the red die is a 1 or 4, the card is either A, 2, 3, 4, 5, or 6. If the red die is a 2 or 5, the card is either 7, 8, 9, 10, J, or Q. If the red die is a 3 or a 6, the card is a K.
If the blue die is 1, the card is either A or 7 or K; if the blue die is 2, the card is either 2 or 8 or K; If the blue die is 3, the card is either 3 or 9 or K; and so on.

When you walk in the room, you can glance at the coin and the two dice and know which card I picked. If you see that the coin is tails-up, the red die is 1-up, and the blue die is 2-up, then you know that the card is black, either spades or hearts, between A and 6, and either 2 or 8 — in other words, the card is the 2 of spades.