Saturday, May 28, 2011

Another card trick

If you liked yesterday's math card trick, you may enjoy this one too, which Jeff Ely just linked over at Cheap Talk.

I enjoy cracking these tricks, and I encourage you to figure this one out before reading further.


OK, here goes. For the record, this was a "1-viewing" trick in my book, whereas yesterday's took me much longer to figure out. (But, you know, I do a lot of this stuff, at least compared to other non-magicians). Today's trick turns on exactly two things. First, you should not be fooled by all the cutting and reassembling of the deck at the beginning of the trick. If you pay attention, the number of cards under each of the 3 aces in the reassembled deck is not altered by the show of cutting and reassembling. (The sequence of moves is equivalent to putting one ace on top of each pile and then stacking the piles). So in fact the position of the aces in the deck is completely and unambiguously determined.

Now that we know the position of the aces in the deck, we could iterate on paper and show that they will survive the up-down-up-down procedure. But that's arithmetic, not problem solving. As problem solvers, we complete our analysis by noting that the up-down-up-down sequence is completely deterministic. It will always result in 3 undiscarded cards, always corresponding to the same positions in the original deck. So to get the 3 aces at the end, all we have to do is put them in the right places at the beginning, whatever those places may be. Since the placement was arbitrary -- set by the magician at the outset -- this dissolves all mystery and completely solves the problem.


Theory versus empirics, once more. The guy in the video evidently knows that this sequence of moves works empirically, but says he has no idea of the underlying theory. He is baffled. And while he can still do the trick, he had to learn a bunch of "arbitrary" steps just right (particularly in the cutting and reassembly). On the other hand, forcing the steps to be consistent with an underlying theory significantly reduces the dimensionality of the information that must be known in order to reproduce the trick. Personally I can't imagine learning how to do this trick in advance of understanding how it works.

I suppose it's an obvious point, in its many forms. Even when an exam seems to test rote memorization -- say by requiring students to reproduce proofs they've seen before -- it is far easier to "memorize" things when you understand the underlying logic. So it can still separate people out on the basis of their understanding. (Of course there can also be a pooling equilibrium where everyone gets everything right, and so forth).


  1. Nice post. I especially like the last two paragraphs about reducing the dimensionality of the information by actually learning something.

    On the pooling equilibrium you mention, the difficulty in distinguishing excellent high-dimensional rote memorization from real learning has always bothered me a little. I'm like you. I have to learn the concept to "memorize."

    The degree to which I agree with you on this point makes me wonder if we're both missing the point and there's a different type of learner out there: someone who has to memorize to learn (to describe that mode of learning in a judgmental way).

    Maybe it makes more sense to draw the distinction as learning taste types rather than learning quality types. Some people can't make sense of a big picture if the details don't make sense while others can't make sense of the details if the big picture doesn't make sense. If a mixture of big picture concepts and mastery of details equals true understanding, both types are getting at true understanding, but the approach is different.

    All of this is to say that I have met people of the other type of learning. They mystify me because my mind doesn't work that way. I also suspect that they spend more time in the process of learning than I do, but who knows? Maybe I have this learning thing wrong.

  2. A thought-provoking comment, Tony. I will have to think more about this.

    In principle I have nothing against rote memorization. Everything we know is a combination of understanding and remembering. Sometimes understanding makes it easier to learn, but other times not. That goes for everyone, to varying degrees, on varying topics.