Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else.
For my third trick, I’ll present something that I first saw when pulling Christmas crackers with my family. I’ll give everyone a piece of paper with six cards printed. I’ll also have a large version of this paper shown at the front of the room (taken from http://diaryofagrumpyteacher.blogspot.com/2014/04/freebie-friday-magic-number-cards.html; see also this Google search if this link somehow goes down):
Here’s the patter:
Think of a number from 0 to 63. Then, on your piece of paper, circle the cards that contain your number. For example, if your number is 15, you’ll need to circle the card in the upper-left because 15 is on that card. You’d have to circle all the cards that contain 15.
Is everyone done? (Points to someone) Which cards did you circle?
At this point, the audience member will say something like “Top left, top middle, and bottom right.” Then I will add the smallest numbers on each card (in this case, 1, 2, and 32) and answer in five seconds or less, “Your number was 35 (or whatever the sum is).” It turns out that the number is always the sum of the smallest numbers on the selected cards.
In yesterday’s post, I gave a similar but utterly unimpressive trick; the trick was unimpressive because it was obvious that the trick used our ordinary base-10 representation of whole numbers. The trick above is much more impressive because it uses binary (base-2) instead of base-10.
The cards above are carefully rigged using binary arithmetic, so that all numbers are written as sums of powers of 2. For example, on the card in the upper left, the first few numbers are
and so on. On the right-hand side, I’ve written each number as the sum of powers of 2 (for the numbers at hand, that means 1, 2, 4, 8, 16, and 32). Notice that each expansion on the right hand side contains a 1. So, if the audience member tells me that her number is on the upper-left card, that tells me that there’s a 1 in the binary representation of her number.
Let’s now take a look at the first few number in the upper-middle card:
and so on. Notice that each expansion on the right hand side contains a 2. So, if the audience member tells me that her number is on the upper-middle card, that tells me that there’s a 2 in the binary representation of her number.
Similarly, the upper-right card has numbers which contain 4 in its binary representation. The lower-left card has numbers containing 8. The lower-middle card has numbers containing 16. And the lower-right card has numbers containing 32. Happily for the magician, each of these numbers is also the smallest number on the card.
So, if the audience member will says “Top left, top middle, and bottom right,” then I know that the binary representation of her number contains 1, 2, and 32. Adding up those numbers, therefore, gives me the original number!