>>9944446I got interested in this a while ago. Typically in high school or college, you can use calculators but in mathematics Olympiads calculators are forbidden so being able to do arithmetic immediately is very important. The book I read was "Secrets of mental math". The core idea is that you should first be able to do one digit sums in milliseconds, and then all basic algorithms boil down to deconstructing operations into a bunch of one digit sums in an efficient way.
There are more fancy algorithms that require you to memorize other stuff. You can safely ignore these. With the basic algorithms, you will already be doing computations much faster than a calculator could and the only limit is how many digits you can store in your brain. The book even mentions that a typical human can keep track of around 8 digits so you can easily do four-digit computations which are good enough for Olympiads. That said, if you want to be like those human calculators you see on TV, you will need to memorize all the special heuristics.
