Long ago on Wrong, But Useful, my co-host @reflectivemaths pointed out the ‘coincidence’ that $7\times8 = 56$ and $12 \times 13 = 156$ - a hundred more.
In fact, it works for any pair of numbers that add up to 15: $x(15-x) = 15x - x^2$, and $(x+5)(20-x) = 100 + 15x - x^2$ - clearly, 100 more. This is a nice, if niche, shortcut - if you’ve got to multiply two numbers that add up to 25 (say, 9 and 16), you can work out the product of the two numbers five less ($4 \times 11 = 44$) and add 100. Much simpler than doing the big sum!
The natural question is, are there any similar patterns? For example, are there pairs of numbers where you can add ten to each and get a number that’s 100 or 200 or 300 more? Let’s find out.
I’m going to assume that the numbers add up to some number $k$, and I’ll let $x(k-x) = N$ for some $x$. Now, let’s add 10 to each: $(x+10)(k+10 - x)$ and see what happens.
Expanding the bracket, I’ve got $10(k+10) + kx - x^2$. However, I know $x(k-x) = N$, so I’ll sub that in: $10(k+10) + N$. To get a number 100 more, $k$ would have to be zero, so that’s not much use. For 200, we could have two numbers that add up to $k = 10$: for instance, $6 \times 4 = 24$ and $16 \times 14 = 224$. It works!
Similarly, for 300, we’d need $k = 20$ - what happens if two numbers add up to 20, such as 9 and 11? Well, $9 \times 11 = 99$ and $19 \times 21 = 399$ - lovely!
What other interesting examples can you come up with?