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:

1. If the coin is heads, the card is red. If the coin is tails, the card is black.

2. 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.

3. 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.

4. 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.