Are there are any pseudo-random number generators that are easy enough to do with mental arithmetic, or mental arithmetic plus counting on your fingers. Obviously this limits to fairly simple math – it needs to be something someone of average mathematical ability can do, or maybe average ability for a programmer, not a math prodigy.
The simplest I have found is the Middle square method, but not only is it known to be a poor source of randomness, it still looks too complex to do without pencil and paper.
If the only way to do this is by limiting the range, like maybe it only can output 8 bit numbers, that is fine. I suspect one of the standard PRNG algorithms would be simple enough in an 8 bit version, but I don’t know enough to simplify any of them from the 32 bit version to an 8 bit version. (All the ones I looked at depend on specially picked seed numbers that are different depending how many bits you are working with, and usually only 32 and 64 bit examples are given.)
A linear feedback shift register is pretty simple, as long as you’re comfortable with thinking in binary (or maybe hex, since it’s easy to map between the two).
A more complex one is Xorshift, but if you know your bitwise operations, it should be quite possible to work with as well.