Number Tricks Explained
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.
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.
Multiplication Race
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