The Goal Is To Astound

Last week I presented a magic trick in which a magician leaves the room while five cards are chosen from a pack by a volunteer. The assistant selects one of the five and gives it to the volunteer. The remaining four cards are placed on a table next to the deck. The magician re-enters, picks up the fan of four cards, and instantly identifies the hidden card. How was it done?

For this trick to work, something about the remaining four cards must carry enough information to allow the magician to deduce the selected card. This is one way. You might think of others.

To determine how much information the stack of four can hold, consider that a normal deck has four suits. Therefore the collection of five initial cards has at least one suit represented by two cards. If a suit is present in more than two cards, there is no difference. The hidden card is selected to be one of the suits with at least two cards in the five. The assistant puts the remaining four cards down with the top card being the same suit as the hidden one. That was easy.

How about the value of the card? That is trickier. Somehow the assistant has to arrange the remaining three cards to convey the value of one of the twelve remaining cards in the selected suit (only twelve since one of the suit is the top card).

Before reading further, try to think of a scheme using three cards that will uniquely code for one of twelve numbers.

If might be possible, but it is not necessary. That is an interesting observation. The coding becomes easier when we realize that we already know one of the cards in the selected suit. Assuming the values of the suit are arranged in a circle from ace to king and then to ace again, by knowing any one card, the hidden card is no more than six steps away from the top card — if one knows which way to go, ascending in values or descending.

So the problem becomes how to code for six values and a direction. The direction is easy. Since the assistant has a choice of which value of the selected suit to hide, they agree ahead of time that the code will count down from the top card value. We now require only a method of coding a number from one to six with the three bottom cards.

Note that we can always arrange the three remaining cards in highest, middle, and smallest value. If two cards have the same numerical value, we include the normal Bridge weighting (i.e. the seven of spades is more valuable than the seven of clubs). Since we have three identifiable objects. They can be ordered in six ways. The magician and assistant agree on some scheme for assigning an order of highest, middle, and lowest value. For instance, highest on top with middle next would be one. Highest on top with lowest next would be two, etc.

Unlike many card tricks, this one can be repeated without danger, but to prevent clever spectators from deducing the method, the basic scheme can be varied simply. For instance, they can alternate increasing or decreasing the counting direction. Similarly the selected suit can be the top card or another location depending on some prior agreement. They can also vary the coding of the six values in some predetermined way. If you try this trick, I suggest keeping it simple and practicing ahead of time with your assistant. The magician should be able to fan the four and not appear to be calculating. The goal is to astound spectators. Any hesitation will alert them to an algorithm in the cards, and then instead of being astounded, they are reduced to trying to guess the scheme.

I selected this trick to feature because it demonstrates that significant information can be conveyed in unsuspected ways. Often much more information is available in nature’s raw data, but we do not know the code. However, unlike the magician and assistant, nature does not change the algorithm to prevent us from decoding her secrets.

Click here to read about my new tutorial on helping seniors. The new version has grown considerably over the original. It has more topics and anecdotes, and fewer typos. While you’re at it, check out my expanded tutorial on decision theory.