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Impressive Math magic with 16 index cards

Here is one of my very favorite Math tricks that's sure to impress your friends (and yourself the first time you try it). I learned this trick over 30 years ago and amazingly enough I still remember it.Binary sorting trick

Take 16 index cards and prepare them as follows:

  1. Make a diagonal cut at the top left of each card as illustrated. This is to keep all the cards oriented the same way because they're going to get mixed up for this trick.
  2. Number each card using the numbers from 0 up to 15.
  3. Using a hole punch cut 4 holes in each card as shown in the card numbered 0 in the illustration. It's important that the holes line up from one card to the next. In other words, when you are holding all 16 cards in your hand you should see four holes that go through the whole set of cards.
  4. Using a pair of scissors cut notches in each of the cards, except for card 0, so that each card matches the corresponding illustration according to is number.

Now for the fun part!

  1. Have the person you're wanting to impress shuffle all the cards so that they're in a random order. Hold the cards face up, with the numbers showing. After shuffling check that the diagonal cut goes all the way through all 16 cards. That's how you know the cards are all oriented the same way.
  2. Take a thin dowel, knitting needle, small screwdriver or other object that can fit through the holes and notches.
  3. Hold the stack of cards in one hand. Put the needle through the rightmost hole (or notch) with your other hand and pull the dowel away from the stack of cards. Some of the cards will come with the dowel - those that have holes in that rightmost hole - and some won't.
  4. Take the cards that came up with the dowel and put them in the front (top) of the stack in the order they came in when the dowel pulled them away.
  5. Proceed to the second hole from the right. Using the dowel again, push it through the hole and pull the dowel away from the stack. Take the cards that came with the dowel (these will have holes in the second-to-right position) and put them on the top of the stack, in the order they came in when the dowel pulled them away.
  6. Repeat with the second-to-left hole and finally with the leftmost hole.

Now, look at the stack of cards. What do you see? They're in order!

Can you explain how this happened?

If you know about computer programming can you relate this trick to the binary number system and to how computers can sort things?

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Comments (17) Trackbacks (2)
  1. In the binary number system, 0 is 0000, 1 is 0001, 2 is 0010, 3 is 0011, ans so on. This is due to base of 2 for binary number or 2 to the power of n, where n is the number of bits with 1. By doing the steps in your post, we are actually doing sorting starting from 2^0. As we move to the next hole, we are moving forward to higher number (2^1)==> (2^2)==>…. It is how computer or digital electronics are handled.

  2. Lim is right.

    This is a sorting process.

    In the end. The cards will sort itself in ascending order. 🙂

  3. Funny, I’m not very good at anything complicated in math but I love math tricks etc.

    I find the way that they teach math in school to be the hard way, my grandmother taught me one or two very basic methods that I still use and never learned in school.

    Do you have anything on your blog that is good to teach kids basic math?

  4. This is, in fact, an old sorting technique used at my local public library in Buffalo, New York in the nineteen-fifties.

    It harkens back to Hollerith (q.v.) techniques used in the late 19th century census.

  5. Okay, I understand exactly what is happening, but just thinking about it is really cool! Neat trick.

  6. It reminds me of the minicomputer. It has spaces for 1, 2, 4, and 8, and can be used to make any number (they use one group of 4 per digit). Your number notches compare to those. It looks fun!

  7. Cool. I can see kids (and many adults) getting a big kick out of this. Think I might have to try it out.

  8. @Lim Ee Hai: The sorting process is not as obvious as you write because if you start with a sorted set of cards, the cards become unsorted before they becomes sorted again. It’s a bit tricky to explain what is going on.

    @Raymond: Yup, they end up in ascending order. The question is ‘why?’

    @Jan: Give me a better idea of what you’re looking for and I can write a post about resources or maybe give you ideas.

    @James: That makes sense. I do believe this technique originated with computer punch cards.

    @Blaine: Thanks. Simple Math can be quite cool!

    @Mallori: Yes, the holes and notches represent binary digits. And, yes, with the binary digits you can combine several to make different numbers. You’re onto it!

    @Jason: Do try it out and let us know how it goes.

  9. The notches represent 1 and the hole represent 0.

    the four holes/notches represent the value place of ones, tens, hundreds and thousands (as in the normal number system that we use) respectively.

    Obviously, the value of 1 is greater than 0 and this process will put greater value below and smaller number above.

  10. I know this post is really old, but I just StumbleUpon’d it, and found it a pretty interesting little trick.

    If someone else falls upon this later and wants to find more information, this is a LSD Radix Sort (I’m pretty sure…)


  11. I just stumbled upon this aswell. really interesting!

  12. Thanks, StumbledApon.

  13. It’s not a trick, but nicely presented. If you’re into a little bit of number theory (or ‘math tricks’) you might want to check out Etheopian Multiplication.

  14. Damn, spelt Ethiopian wrong!

  15. its very fun to try where do you get your idea of that amazing math solving?

  16. I did this “trick” when i was 8yrs old.

  17. its too interesting.my frns appreciated me.thanks a lot……..

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