(This was suggested to me by a friend.)
Find the prime factorisation of 12318876808768112319.
Factorisation puzzles are familiar, and obviously the point is to minimise the use of computer assistance, by making ‘clever’ observations.
Once you see how to do it, you’ll notice that there is a still a non-trivial amount of checking to do. Thus, a meta-puzzle suggests itself: to find a natural number N, so that the problem of factorising N by hand contains lots of interesting and clever observations, and very little checking.
I don’t know how to solve this meta-puzzle!