1089 and all that

An extract from a new book written by David Acheson.

1089 and All That

Think of a three-figure number.

Any three-figure number will do, so long as the first and last figures differ by 2 or more. Now reverse it, and substract the smaller number form the larger. So, for example,

782 - 287 = 495.

Finally, reverse the new three-figure number, and add:

495 + 594 = 1089.

At the end of this  procedure, then, we have a final answer of 1089, though we have to expect, surely, that this final answer will depend on which three-figure number we start with. But it doesn't. The final answer always turns out to be 1089.

Why?

The first step, if you recall, is to take a 3-digit number, reverse it, and subtract the smaller from the larger.

Suppose, then, that the larger of the two numbers has digits a, b, c; then its actual value is 100a + 10b + c, and after 'reversing' and subtracting we will have 100a + 10b + c - (100c + 10b + a), which is the same as:

100a + 10b + c - 100c - 10b - a  = 99a - 99c = 99 (a - c).

As a and c are whole numbers, this shows, then, that at the end of the first part of the trick we will always end up with a multiple of 99.

Now, the 3-digit multiples of 99 are 198, 297, 396, 495, 594, 693, 792, 891, and we note at once how the first and third digits of each of these add up to 9. So, when we reverse any one of these numbers and add - which is the last part of the 'trick' - we get 9 lots of 100 from the first digits, 9 lots of 1 from the third digits, and 2 lots of 90 from the second digits, giving

900 + 9 + 180 = 1089.

So we have done a little mathematical conjuring, and a bit of algebra helped us along the way.

You can buy this book from Amazon.co.uk or Amazon.com.