The final installment, the big reveal: why does the ninja trick of multiplying by 77 and finding the nine’s complement work? My friend, that is an excellent question.

The reason is this: $77 \times 13 = 1,001$. And it turns out, 1001ths are not all that hard to work out. (This is, in fact, just an extended version of the ‘adjusting fractions’ version of a previous ninja secret).

To work out $\frac{539}{1001}$, you might start by saying: that’s 0.539 (less about a thousandth). A thousandth of 0.539 is 0.000539, which — if you take it away — leaves you with 0.538461. If you do it by hand, you even see where the dropping one and using nines come from.

Even that adjustment needs adjusting - this is about one part in a million too small, which is why it recurs.

As tricks go, this is currently one of my favourites.


* Edited 2017-04-07 to add links](