I've seen various versions of the above image floating around social media on how to multiply large numbers in your head. I know it doesn't work for all numbers, but why does it work for some numbers and what is the rule for knowing whether it will work or not for a certain pair of numbers? A brief Google search didn't pull anything up for me, but perhaps I missed something. I tried multiplying 97 and 96, 98 and 99, 90 and 90, 89 and 89, 89 and 87, and then I tried 50 and 50, the first for which it didn't work for me, and I could not figure out why. Thanks so much!
$(A-B)(A-C)\;=\;A^2-AB-AC+BC\;=\;A(A-(B+C))+BC.$
Therefore, with $A=10^2$ and $B=3$ and $C=4$, we have $$(97)(96)=(10^2-3)(10^2-4)=10^2(10^2-(3+4))+3\cdot 4= (100)(93) +12.$$
Another example: $(91)(92)=10^2(10^2-(9+8))+9\cdot8=8300 +72.$