I can use a few tricks, like using logarithms, or near neighbors to rapidly get very good approximations, but that's it.

I remember hearing that Meyer Lansky, as a child, would watch freight trains go by and add up their serial numbers as they passed. One of the things that helped him in his criminal enterprises is that he never needed to write anything down. He could memorize all of the accounting entries for his illicit businesses.

In high school, we used to play games like finding logarithms or cube roots of randomly chosen numbers, and the first to get 4 sig figs right was the winner. Also, if no one got a correct solution in 60 seconds, that's when we stopped.

Some of the commonly used tricks were : knowing and using logarithms of common numbers (2,3,5,7), using binomial expansions and nearly-linear interpolations between nearby outputs. The art of interpolating along the logarithmic curve (make a linear interpolation and adjust upwards correctly) usually helped the most.

I can only multiply 2 digit numbers and some 3 digit numbers (by stricly multiplying), but there are always nicer big numbers, and often, you don't need the exact product.

PS : Ypu might want to Google "Shakuntala Devi" - she can consistently multiply pairs of 10 digit numbers in under a minute. Her record, I think, is a pair of 13 digit numbers in under 30 seconds !

I can multiply or square 2 digit numbers very quickly. With that, I can divide or find square roots for 4 digit numbers to 2 digits and can usually interpolate to extend the precision to 3 or 4 digits. The only work I've done with logs is with decibels. With dB's, if you know the log of 2 is about .3 and the log of 10 is 1, 100 is 2, etc, you're accurate enough for most situations (i.e. - If you have an input of 25 Watts with a 17 dB gain, your output is about 2500/2 or about 1250 Watts).

Multiplying long numbers together can be done with the following method:

Units = Units x Units
Tens = Units x Tens + Tens x Units
Hundreds = Units x Hundreds + Tens x Tens + Hundreds x Units
Thousands = Units x Thousands + Tens x Hundreds + Hundreds x Tens +Thousands x Units

and so on. Of course you have a carry term from each stage. Keeping track of it all isn't that difficult. Speed and accuracy are more of a problem, but I dare say they would come with practice.

I feel a bit like that too. I didn't spend much time memorizing multiplication tables as a child, and I'm pretty poor at mental arithmetic. I can usually work algebra problems through many steps in my head, though.