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## Math Puzzles: Magic 9

By ItsNotMagicItsScience
Image Source:
http://www.sxc.hu/photo/1117094

From the previous math tricks involving lightning quick mental additions, this one involves a calculator, and some napkins – to catch the drool from your audience's mouths! It starts off simple: without you looking, ask a friend to take a random single-digit number and multiply it with another random single-digit number, until they reach a 6- or 7-digit number (for example, 4 x 6 x 9 x 6 x 2 x 3 x 8 x 9 x 1 x 4 x 5 x …. ).

From the final number ask them to choose one secret digit that they would not reveal to anyone, and tell you the remaining digits, in any sequence they choose. For example, if the final number is 282,240, let's say they take 4 as the secret digit. After that, they would let you know the remaining digits in any sequence, eg., 2,2,8,0,2.

Amazingly, you could tell them what the missing number is, despite having a random selection of numbers that no one could predict.

The secret: Given such a long sequence of single-digit multiplication, it's highly likely that the number is a multiple of 9. We know this if the sum of the individual digits in the final number is a multiple of 9. For example, in the number 282,240.

2+8+2+2+4+0 = 18

which is a multiple of 9.

Knowing this, if someone takes away the secret digit from the number (4), what you're left is:

2 + 8 + 2 + 2 + 0 = 14

It doesn't matter what sequence the digits are given to you – adding to the allure of the trick. It could be

8 + 2 + 0 + 2 + 2 = 14

And so on.

To find the missing digit, find the remainder needed so that the sum of digits would be the nearest multiple of 9 – which is 4! Try another number, this time in the millions: 18,370,800.

Let's say a friend takes 7 as the secret digit. All that's revealed to you is: 1,8,3,0,8,0,0

1+8+3+0+8+0+0 = 20

What's needed to make it up to the nearest multiple of 9? You got it... 7!

Caveats:
There are two places where this trick could go wrong.

The number at the end of the series of multiplication is not a multiple of 9. This can happen when, within the multiplication sequence, your friend doesn't use 9, or a pair of 3s and 6s. Given the long string of numbers, it's highly unlikely, but if it does happen, just ask your friend to reverse their number and then subtract their original number. This will instantly give you a multiple of 9 and you can try again.

For example:

3,010,560, which gives you:

3+0+1+0+5+6+0 = 15

That's not a multiple of 9.

To make it a multiple of 9, reverse their number and then subtract their original number:

3,010,560 – 650,103 = 2,360,457

Adding up the individual digits gives you:

2+3+6+0+4+5+7 = 27

And do the trick.

What happens if the secret digit is 0 or 9? This a pickle. In the example above of 18,370,800, let's say your friend took one of the zeros as a secret digit, leaving you with:

1+8+3+0+8+7+0 = 27

Now you're stuck with a dilemma. You can't tell if the chosen digit was a 0 (to make a digital sum of 27) or a 9 (to make a digital sum of 36). This is where you have to ask your friend for one more clue: Is their number is even? If the answer is “no”, then they chose the number 9, if they said it was “even” – or looked confused – then it's 0.

This trick was devised by James Grime, a faculty member in the Applied Mathematics and Theoretical Physics department in Cambridge University. This particular trick was made possible thanks to what is called modular arithmetic, a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus.

Want to know more about modular arithmetic? Delve deeper into James Grimes's explanation of the trick here.