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06/20/2004: "Binary Math Trick"
Prompted by Think Again! and a couple of eight year olds: my favorite math trick.
It involves a number of tables; when I first saw it, each table was on a separate card. One set of tables:
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To work the trick, the 'mark' is to pick a number from 1 to 31. Then, they say which tables contain that number. ("Is it on this table?" Yes. "Is it on this table?" No. "Is it on this table?" Yes. "Is it on this table?" Yes. "Is it on this table?" No.) After finishing, the 'Magician' confidently announces the number. (In the above example, if the mark's number was on the tables which contained the numbers 1, 4, and 8 but not on the tables which contain 2 and 16, the mark's number would be 13.)
The key to understanding the trick is binary, or base-2 math. Base-10 math is based on our hands. When we write 11. we mean 10, the number of fingers we have, plus 1. "123" is base-10 is 1*10*10 + 2*10 + 3.
We wouldn't write 123 in binary though. When we add 1 to the largest digit in base 10, 9, we reset the 9 to 0 and carry the one to the next column (9+1=10). Likewise, when we add 1 to the largest digit in base 2, 1, we reset the 1 to 0 and carry the one to the next column (1+1=10). Counting from one to sixteen in binary: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000. If we were to count to 123 in binary, by the time we got done carrying the ones, we'd have 1111011.
Anyway, knowing how binary works lets us describe the numbers on the individual cards. The first card can be described as XXXX1. That is, all the numbers on the first card can be expressed as five digit binary numbers. If number appears on the first card, the last digit of the number expressed in binary is 1 (it is an odd number). As it happens, all odd numbers appear on the first card.
The second card can be described as XXX1X; the third card, XX1XX; the fourth card, X1XXX; and the fifth card as 1XXXX.
So when the 'magician' asked the 'mark' if the number was on cards 1, 2, 3, 4, and 5; the magician was really asking the mark to express the number in binary. In the above example, "YNYYN" read backwards becomes "NYYNY"; replacing yesses and noes with ones and zeroes, we get 01101, the binary representation of 13.
When I teach this trick to 8 year olds, I first make up some cards. Then I do the trick. Then I teach them the trick. I tell them to keep a running total of the numbers in the upper left hand corners of the cards which have a 'mark's particular number. Or, in the above example, "Is your number here?" "Yes" Upper Left:1. Running total: 1. "Is your number here?" "No" Running total: 1. "Is your number here?" "Yes" Upper Left:4. Running total: 5. "Is your number here?" "Yes" Upper Left:8. Running total: 13. "Is your number here?" "No" Final answer: 13.
I like this trick for kids for three reasons. First, they like it; they can mystify their friends. Second, it gets them doing math. Third, it plants an 'aha' moment; at some point in the future, they will come to understand binary math and say "aha".
