Mental arithmetic is fun. It has some use, yes. It’s always nice when you’re doing work to have some idea what a reasonable answer looks like. But mostly it’s fun to be able to spot, oh, 24 times 16, that’s got to be a little under 400.
I ran across this post, by Math1089, with a neat trick for certain multiplications. It’s limited in scope. Most mental-arithmetic tricks are; they have certain problems they do well and you need to remember a grab bag that covers enough to be useful. Here, the case is multiplying two numbers that start the same way, and whose ends are complements. That is, the ends add together to 10. (Or, to 100, or 1000, or some other power of two.) So, for example, you could use this trick to multiply together 41 and 49, or 64 and 66. (Or, if you needed, to multiply 2038 by 2062.)
It won’t directly solve 41 times 39, though, nor 64 times 65. But you can hack it together. 64 times 65 is 64 times 66 — you have a trick for that — minus 64. 41 times 39 is tougher, but, it’s 41 times 49 minus 41 times 10. 41 times 10 is easy to do. This is what I mean by learning a grab bag of tricks. You won’t outpace someone who has their calculator out and ready to go. But you might outpace someone who has to get their calculator out, and you’ll certainly impress them.
So it’s clever, and not hard to learn. If you feel like testing your high-school algebra prowess you can even work out why this trick works, and why it has the limits it does.
nebusresearch 
I’ve been running the 16×24 problem through my students, mainly to see how they do it. Me, I go for “lots of doubling” which gets me the answer nice and fast.
SOMEONE (glares at upstairs) had the answer in about 3 seconds by using the difference of two squares. Nerd.
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Oh, yes, difference of squares is a really good way of doing that. I almost never think to use it, though, for some reason.
(For those who don’t see it, the difference of squares comes from binomials: (a + b) times (a – b) is equal to a^2 – b^2. Really helps with certain problems.)
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