A card trick

Yesterday evening, a friend of my mother’s regaled us with a card trick. (Fair warning, this post will reveal how the card trick works. But it’s not an especially amazing trick. I don’t think my revealing it will be a great loss to the magic profession.) He put a deck of cards in his back pocket, then asked my fiancée to pick two suits, then to pick her favorite of those, then to pick a run of five cards in the suit, then to pick another entirely different run of five cards, then to pick three of those, then two of those remaining three, then one of those remaining two, which were the two and queen of hearts. She picked the two of hearts, and he said, 'That leaves the queen of hearts as the unnamed card. Pick any number up to 52, and I will produce the queen of hearts from my back pocket in exactly that number of tries.' My fiancée, feeling a bit impish under the influence of Glenlivet, chose 51. (Actually, her first choice was zero, which might be interesting to a linguist studying the meaning of "up to 52", but since he clearly wouldn’t be able to produce the queen of hearts in zero tries, he made her pick again.) So the dilettante magician pulled one card after another out of his back pocket, until impatience set in, then he pulled most of the deck out of his pocket, counting out the cards. At about 40, he exhausted the ones in his hand and went back to his pocket. At 51, he pulled out the queen of hearts. We applauded him. How could he have done it? It might seem obvious to the reader. Sitting at the table last night, the explanation wasn’t immediately apparent. But some reflection revealed the answer.

Got it?

The answer got me thinking about magic tricks: Do they all turn on some kind of logical contradiction? The magician makes you believe a proposition that contradicts another proposition that would explain the trick. Due to the contradiction, you reject the explanatory proposition. But now, given everything you know about the immediate context and the world, the magician should not be able to do what he did. The conclusion of the trick is a physical impossibility. That’s why we call it magic, right? Now, if you don’t believe that magicians are people who can break the physical laws of our universe, and if the propositions you believe lead you to the conclusion that the magician broke the physical laws of our universe, then it must be the case that the proposition that contradicts the explanatory proposition is actually false, and that the explanatory proposition is true. 

Let’s take the card trick above as an example. It would seem that the only way for a person to avoid producing the relevant card until they reached the randomly chosen number would be to know where the relevant card was in the deck before placing the deck in their pocket. But we also know that the participant from the audience chose the relevant card after the magician put the deck in his pocket. Let’s set these two propositions apart.

(1) The magician knew the location of the relevant card before placing the deck in his pocket. (the explanatory proposition)

(2) The relevant card was not chosen until after the magician placed the deck in his pocket. (the magician's bait)

Clearly propositions (1) and (2) cannot simultaneously be true (assuming he didn't memorize the position of every card before placing the deck in his pocket).  So we have to jettison one of them. Everyone in attendance saw the relevant card get named. It happened after the deck was in his pocket. So (2) has to be true. Therefore, (1) cannot be true. The magician was able to find the relevant card in some other way. But how? It's this question that produces wonder in the audience. We've rejected one of the contradiction-creating propositions, but what we're left with is an unexplainable result…unless of course you believe in magic. ESP, say. Some sort of paranormal intuition. The magician was able to know, in an extrasensory way, where the queen of hearts was in the deck.

But no one in our society seems to really believe that card trick performing magicians use ESP. Perhaps we have seen too many card tricks explained, or performed badly by friends. (Psychics are a whole different story. Lots of people in our contemporary society seem happy to believe in the paranormal abilities of psychics. But if some people really do have such powers, why isn't there any scientific documentation of it? Why don’t these psychics use their power for something more than charging a room full of believers forty dollars a head for cryptic messages from dead relatives? Oh, and don't forget to buy a signed copy of the autobiography at the merch table on your way out! No, the feats of psychics must work similarly to those of card trick magicians.) So if ESP is out, then how did my mother's friend find the queen of hearts in his back pocket on the 51st pull?

Maybe we rejected the wrong proposition. Let's reject (2), and leave (1) intact. The relevant card was chosen before the magician put the deck in his pocket, and moreover he knew where in the deck it was. Perhaps the explanation was clear to you from the beginning; after all, my mom's poor friend didn't have much luck with my fiancée. He had to ask her to pick a second run of five cards. Then when she chose the two of hearts, he had to resort to making the unnamed card the chosen one, which is a pretty fishy way of having a participant choose a card! The trick wasn't in magically, via a sixth sense, finding the queen of hearts in his back pocket on the 51st pull. The trick was in convincing the audience that the relevant card was chosen at random by my fiancée. It's easy to find the queen of hearts in your back pocket if you know ahead of time that that is the card you'll be looking for and if you put it on the top of the deck.

The trick seems to be in getting the audience to reject the explanatory proposition, but that proposition can never be completely rejected. The audience rejects (1), but they also know that (1) is the only non-magical explanation of how a human being can pull a specific card out of his back pocket at a specific point in a sequence. So we reject (1) and we're left with magic as the only explanation. Since the audience doesn't believe in magic, they know they've been tricked but they don't know how. Even so, the result is impressive.

I imagine professional magicians could spice this trick up a little more. Before putting the deck in their pocket, they'd shuffle it up a bit. Masters of sleight of hand, they'd be sure to keep the queen of hearts on top or on bottom the whole time. They might let the participant cut the deck, but again, they'd put the queen of hearts on bottom. Or maybe the queen of hearts is already in their pocket, and they just keep track of where it is when they stick the deck in there. In "Intuition Pumps and Other Tools For Thinking", Daniel Dennett writes that card magicians keep a few tricks lined up at all times. They start off by trying to do a hard one, one that the percentages say they are unlikely to pull off. If they realize they can't pull it off, they try for an easier one. If that fails, they move to an even easier one and so on, until one works. If they are forced to their weakest trick, then the result will seem less impressive, but the audience will still be pleased. My mom's friend mentioned that if he performs the trick with a female participant, the relevant card is always the queen of hearts. If it's a male participant, it's always the ace of spades. Chances are better that members of each respective gender will home in on each of these respective cards. If my fiancée had been typical, he might have gotten her to pick the queen of hearts herself and quickly, convincing the audience even more steadfastly that (2) is true.    

To me the interesting question now is, do all magic tricks turn on this kind of contradiction? Is the audience always faced with a set of contradictory propositions, one of which they must reject? And is the magician's job simply to convince the audience to reject the wrong proposition? To reject the proposition that will force them to posit magic in order to explain the result?