Past Time For A Magic Trick

After several columns looking at problems with real life situations such as political hit pieces, it is past time for a magic trick. This one, like all good tricks, appears in many variations, and I do not know where it originated. If anyone knows its history, tell me, and I will give it credit. Several variations exist without due credit given to whoever thought it up. My variation is only a retelling in my own words. Since I am not a sworn magician (anymore), I can safely reveal the answer next week. Actually I will reveal at least one answer. As you will see, this trick has the potential of more than one explanation.

A magician hands a standard unmarked deck without jokers to his assistant with instructions to have a member of the audience select five cards at random. Then the assistant will look at the cards, show them to the audience, and select one of them to hand back to the volunteer. The assistant will then place the remaining four cards on a table next to the stack of 47 unselected cards. After giving these instructions, the magician leaves the room so the process can proceed in secret.

The idea here is to divide the randomness of card selection up between two people. The assistant has no control over the first filter as the volunteer selects five cards, and the volunteer has no control over the second filter as the assistant reduces that pool to a single card.

When all this is done, the magician returns. He looks at both piles, but does not disturb the deck. He picks up the stack of four, fans it out, and announces that one card is missing. He names the missing card (suit and value), and it is indeed the one the volunteer is holding. How did he do it?

There are obviously many ways to do this trick, but we require no communication between the assistant and the magician such as secret signs or key words. In fact, the assistant does not have to be present when the magician re-enters the presentation room. Also, the magician was truly separated from the action. That is, he had no electronic monitoring equipment, spy holes, or other means of observing the process of card selection.

First hint: since this is a column about decision theory, statistics and probability, you can assume that some rational thinking is necessary here about how the magician can look at the two piles, fan out the four, and immediately know the chosen card. The method has to be simple enough that it does not require pencil and paper or a long time to calculate because that would ruin the effect.

Second hint: The trick can be repeated, but if it is repeated too often, an astute observer can figure it out. With a slight modification of the mechanism, this situation can be avoided. However, even with reasonable modifications, a programmed computer watching a large number of examples could probably decipher the method.

How is it done?

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.