Your mum or dad may have taught you this when you were a child: You take a small sharp knife and try to get through the cooked potato, if you can do it easily, it's cooked. It always works of course, but it's a bit subjective, "easily" is not very easy to appreciate and then it depends on the variety of the potato, "Rattes" for example are much firmer than the "Belles de Fontenay".
Can you do any better than cutting through them with a knife?Yes, and it's quite simple, let's imagine you want to make melting potatoes, and therefore be sure they are well cooked, then in that case, stick the famous knife in the potato, vertically, and lift the potato.
It comes with the knife? => undercooked, continue cooking
Does it fall off the knife by itself, or does the knife come out by itself? => they are cooked!
Obviously this only works with good sized potatoes, with grits or rattes it's wrong, the weight is too low for them to fall back...
In summary: To test the cooking of a potato, stick a knife vertically into it and lift it up, if the potato stays in place or falls back on its own after a few seconds, it is well cooked.