A Young Mathemagician October 30, 2011

A Young Mathemagician

By Hemant Mehta
October 30, 2011

Ethan Brown is only 12, but his ability to do tricks using mental math is incredible.

At last week’s Skepticamp in New Hampshire, Ethan created a 4 x 4 magic square after audience members filled in some of the boxes for him at random. And he even found a way to get the columns and rows to add up to an appropriate number for that particular audience 🙂

When I watched that, I was thinking he’s like a younger Art Benjamin… and then I saw this video on Ethan’s website:

…. and I’m jealous. I want his skillz.

Ethan, when you come to Chicago next month, you have an open invitation to perform in front of all my classes 🙂

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  • Skeptic

    You know this a magic trick right?  you do not need savant level math skills.  Just need to know the method and be able to keep it going.

  • Of course it is, but if you watch the videos, it’s not all magic tricks anyone could perform. You need to be able to do a certain amount of tough mental math in your head.

  • He could still be a creationist.

  • There’s another issue here that’s worth mentioning: Not only is he doing tough arithmetic and keeping a lot of digits in his head, but he’s a really good presenter. He has serious stage presence and knows how to work his audience. Quite impressive. 

  • That seems a bit unnecessarily snarky.  

  • This kid is pretty great, thanks for sharing.

  • Pollracker

    Let’s get this kid working on quantum theory

  • Meettheskeptics

    He’s my son and I assure you he is not a creationist.

  • Rieux

    How about that calculation, at 2:45 of the video, of the number of possible permutations that are available for three volunteers each successively entering a (pseudo)random 1-to-20 number in a (pseudo)random cell in the 4 x 4 grid?

    By a very back-of-the-envelope, first-order calculation, that looks to me like (20 x 16) x (20 x 15) x (20 x 14) possible permutations—which comes out to nearly 26.9 million, not 4.48 million.

    Is Ethan excluding (arguably redundant) rotations? I.e., is he presuming that

    1 X X 2
    X X X X
    X X X X
    X X X 3

    …is equivalent to , and therefore not an independent possibility from,

    X X X 1
    X X X X
    X X X X
    3 X X 2

    …(as well as two other rotations of the same case)? And then I suppose there are inversions, like this:

    1 X X X
    X X X X
    X X X X
    2 X X 3

    Presumably that explains the difference between the simple-multiplication estimate and the lower number he provided. It may be beyond my math chops to find the latter in precise terms, though. (I’d probably just divide 26.88 million by 8 (original case + 3 rotations + 4 inverted rotations) and come out with 3.36 million, which is still not the number Ethan provided.)

  • Rieux

    …thinking about this a little more, I suppose there are cases that don’t have 7 other equivalent permutations. Such as this:

    1 X X X
    X 2 X X
    X X 3 X
    X X X X

    You can rotate that (creating four permutations from the one), but you can’t really invert it (creating four more) without making yourself a non-equivalent case.

    Which makes me suspect 4.48 million is correct. The math to get there is over my head, though.

  • Meetheskeptics

    Actually, while most Magic Squares do fall pretty solidly into the “trick” category, this one actually takes some pretty tough math. He purposely set out to create one that not just anyone could do, and the 3 random squares with the 3 random numbers was the twist that took it out of the “trick” category and made it more of a puzzle. I could definitely be wrong, but I have yet to see another presentation of the Magic Square that allows free choice like that.You are definitely correct though that you don’t need savant level skills. You need a certain level of competence and a passion for the subject that motivates you to practice. Ethan just loves Math & is good at it but more importantly, he practices a lot. Practice anything enough & you’ll get good at it.

  • Meettheskeptics

    Hi Rieux,

    You are zeroing in on it. The original formula you proposed is close but is missing one crucial step. You need to divide by 3! (or 6). Here’s why.The 3 volunteers in this case filled it in as follows:

    X X X X
    X 18 X X
    X X X 13
    X X 5 X

    Between the 3 volunteers, there are six different ways this exact result could have come about.

    1) V1 = 18  ,  V2 = 13  ,  V3 = 52) V1 = 18  ,  V2 = 5  ,  V3 = 13
    3) V1 = 13  ,  V2 = 18  ,  V3 = 5
    4) V1 = 13  ,  V2 = 5  ,  V3 = 18
    5) V1 = 5  ,  V2 = 18  ,  V3 = 13
    6) V1 = 5  ,  V2 = 13  ,  V3 = 18

    By your original formula, this would be counted as 6 different outcomes, when in reality, they are all just one since it really doesn’t matter who put which number where. For that reason, you need to divide the 26,880,000 by 6, which gets you to the 4.48 million.

  • Rieux

    Thanks for the response—and the explanation!

    But… you didn’t take the rotation/inversion issue into account?
    Then tsk, tsk! The case on Ethan’s pad is also equivalent to (i.e., one can use the same arrangement of 13 other numbers, just slightly rotated and/or inverted, to complete the same magic square in) each of these:

    X X X X
    X X 18 X
    5 X X X
    X 13 X X

    X 5 X X13 X X XX X 18 XX X X X

    X X 13 XX X X 5X 18 X XX X X X

    X X X XX 18 X XX X X 5X X 13 X

    X X X XX X 18 X13 X X XX 5 X X

    X 13 X X5 X X XX X 18 XX X X X

    X X 5 XX X X 13X 18 X XX X X X

    Given that, it sure seems that the 4.48 million number needs to be divided again, by some number between 4 and 8. Heck, I bet there are fewer than a million truly independent possibilities!

    Now I’m starting to suspect that Ethan really did memorize all the relevant solutions…. 😉

  • Rieux

    Oh, for @#$#@% sake. Disqus, I hate you.

  • Meettheskeptics

    That’s definitely a good point and one he hadn’t considered. My gut tells me that you’d simply need to divide by 4 as you could rotate each outcome 4 ways & technically it’d be the same, just turned sideways. That said, I do know that he wouldn’t approach:

    X X 12 X
    X 5 X X
    X X X 17
    X X X X

    the same as….

    X X X X
    X X 5 X
    X X X 12
    X 17 X X

    even though they could technically be viewed as the same. SO, depending on whether or not you view those as the same or different (and I think an argument could be made either way), you either have 4.48 million or 1.12 million possibilities. Good points!