Number Tricks Explained

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Addition Race

Choose the last three numbers so that they pair up with the given three and make 3 x 99 or 297. If the six are four-digit numbers, the magic total is 3 x 9999 or 29,997.

The magic total for ten-digit numbers is 29,999,999,997 but getting that in an instant might make your audience a bit suspicious.

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Subtraction Secret Digit

This trick involves the digit sum which contributes a characteristic value to any number. For example the chosen 7832 has a digit sum of 20, and the rest of its value, 7812, is all groups of nine. Scrambling to get 3287 we have a number that is also 20 plus "a bunch of nines." How come?

7832 = (7 * 1000) + (8 * 100) + (3 * 10) + 2 = 7 + 8 + 3 + 2 + (7 * 999) + (8 * 99) + (3 * 9)

3287 = 3000 + 200 + 80 + 7 = 3 + 2 + 8 + 7 + (3 * 999) + (2 * 99) + (8 * 9)

When we deduct 3287, the 20 goes away and we are left with only groups of nine. It doesn't matter how many – any multiple of nine will have a digit sum which is also a multiple of nine, and that's why the trickster can guess the secret digit that makes the total come out right.

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Multiplication Race

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This trick takes advantage of the algebraic fact that (x + y)(x – y) = x – y . Here is a picture of one example. (20 * 20) – (4 * 4) = 400 – 16 = 384

384 = (20 + 4)(20 – 4).

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Division – Out and Back

When the three digits are repeated it has the effect of multiplication by 1001, for example

321 * 1001 = 321,321. Since 1001 is 7 * 11 * 13, the six-digit number is divisible by those factors, and upon completion of the three divisions, the original number re-appears.

321,321 / 1001 = 321

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