Math and "Mathemagic"

Arthur Benjamin has been a long time hero of mine, ever since I first saw his TED Lecture on how he does mental math. Today, I managed to convince my science teacher to let me skip some class and see Benjamin speak on campus at the High School. Much of his talk was extremely similar to his lecture at TED, sometimes even word for word, but this time he had some very cool never-before-seen tricks.

For his first trick (if you can really call it that), Benjamin had a student come on stage where he produced an imaginary deck of cards. He asked the student to pick an "invisible" card from anywhere in the deck. This seemed strange, as there were really no cards at all, however the student pretended to take a card, and then to flip it up-side down and place it back in the imaginary deck. Benjamin proceeded to pretend to place the cards in his pocket, then withdrew a real deck of cards. Benjamin fanned them out and pointed out that one of the cards was facing backwards, as if the student had placed it there. He then asked the boy what card he imagined he was holding when he took the pretend card. "A seven of hearts" was the student's response.

We were all skeptical as to what was going to happen next, but it really was remarkable what he was able to do. He pulled out the card and showed it to the crowd, a seven of hearts.

I was blown away to say the least. Arthur Benjamin seemed to know ahead of time what card would be in the student's mind, as the real cards he took out of his pocket were arranged neatly inside a box. One theory is that Benjamin used simple probability to accurately guess what card the student was going to imagine. Seven is often times what one thinks of when asked to choose a number between one and ten. Perhaps with cards, people had the tendency to usually pick a seven of hearts. However, this would leave a large margin of error as the volunteer could be thinking of a different card just as easily, making it only probable that Benjamin would be correct.
However, he is a mathematician, so there may be some way to calculate what card the person would be thinking of when pulling a card out of a certain part of the imaginary deck. For instance, if the student had pretended to pull out a lower card, then he may subconsciously be thinking that the card he is choosing is a King or Queen. However, this too leaves a large margin of error, as the volunteer could be thinking of an ace of clubs while imagining to pull a card from the top half of the deck.

To be honest, this one really stumped me. There are a number of theories, most of them are possible, but not necessarily probable.

I was amazed by Arthur Benjamin's math abilities, so much so that hours later I bought his book, Secrets of Mental Math. I don't think his card trick will be in there, but I will hopefully have a more interesting concept of math when I finish reading it. And hey, maybe I'll finally be able to square three digit numbers in my head...