Monday, June 17, 2019 0

Start by writing the number 1089 on a piece of paper and put it in your pocket.

Now get someone to choose a three digit number where the last digit is at least 2 less than the first digit. Turn it round to form a new three digit number, then subtract it from the first one.

For example, 481 − 184 = 297.

Now turn the new number round to form a fourth three digit number, and add it to the third.

297 + 792 = 1089.

Produce the bit of paper from your pocket!

Of course, the result is always 1089, but why? Consider the number 481 which is actually 4 × 100 + 8 × 10 + 1. Let’s generalise that to the number abc which is actually 100a + 10b + c. Turned round, cba becomes 100c + 10b + a. Now we need to subtract:

(100a + 10b + c) − (100c + 10b + a) = 99(a c)

Now you have to follow the next steps carefully: we have to rewrite this in the form of a reversible three digit number:

99(a − c) = 99[(a − c) − 1] + 99
= 100[(a − c) − 1] − 1[(a − c) − 1 ] + (90 + 9)

We’re nearly there: let’s swap the last two terms:

= 100[(a − c) − 1] + (90 + 9) − 1[(a − c) − 1]

This would be in the form of a thee digit number if the last term had a plus sign, so:

= 100[(a − c) − 1] + (10 × 9) + 1 × [10 − (a − c)]

So we now have a number with digits [(a − c) − 1], 9, [10 − (a − c)]. We turn these round, to get: 100[10 − (a − c)] + (10 × 9) + [10 − (a − c)]

Now add these numbers to get:
101[(a − c) − 1] + (20 × 9) + 101[10 − (a − c)] and if you simplify this you will get 1089.

Phew!

1089 is itself quite an interesting number. For instance, use your GDC to multiply it by the list {1,2,3,4,5,6,7,8,9}, and you will see a pattern in the results. And of these results, 1089 × 9 = 9801. Furthermore, in another nice twist, 1089 = 332 = 652 − 562 !

When we get interesting results such as these, the mathematician naturally tries to find extensions: this is the way new discoveries are often made. For example, I’ve just tried the 1089 trick using 4 digit numbers. So far, I seem to have come up with two final answers: 9999 or 10890. And the next obvious question? What properties of different 4 digit numbers lead to the different results? At the moment I genuinely don’t know, having only tried a few numbers – perhaps, you could have a look for me.