Does really everyone think I'm some sort of magic maths wizard who answers all your funny maths questions?
Earlier this year, my new class team colleague instead of introducing me the new class, asked me "How much is 79 x 8 - 16 x 7 without paper or calculator?"
"Eh, 520, I'm Veronika, nice to meet you too."

Of course, I like such questions, even though it's a bit odd instead of greeting, but I've earned the respect I guess =)
So how did I do it? By decomposing those numbers into nicer ones before multiplying. All could be done in your head, really:

Step one: 79 is 80 - 1, so I multiply 80 x 8, which is like 8 x 8 x 10 = 64 x 10 = 640 and subtract the "one 8 on top", therefore I have 640 - 8 = 632. I visualise this number and keep it back in my head...
Written in "simple" maths: 79 x 8 = (80 - 1) x 8 = 80 x 8 - 1 x 8 = 10 x (8 x 8) - 1 x 8 = 10 x 64 - 8 = 640 - 8 = 632

Step two: 16 is 15 + 1, 15 is 10 + 5, which is good as 5 is half of ten and multiplying by ten is always easy as same as finding a half of (almost) any number. So I have 10 + 5 + 1 = 16, multiplying all by 7 will give me 70 + 35 + 7 = (with a little reorganising) 70 + 30 + 5 + 7 = 100 + 12 = 112.
Written in "simple" maths: 16 x 7 = (10 + 5 + 1) x 7 = 10 x 7 + 5 x 7 + 1 x 7 = (or 10 x 7 + (10 x 7) / 2 + 1 x 7 = 70 + 70 / 2 + 7 =) 70 + 35 + 7 = 70 + 30 + 5 + 7 = 100 + 12 = 112

Step three: Combine everything together - dig out the first number from the back of your head... remember it was 640 minus 8 -> 632 and the second one was 100 + 12 -> 112... Original question was to subtract those two, therefore 632 - 112 = 600 - 100 + 32 - 12 = 500 + 20 = 520.

...nice to meet you... =)