my favorite answer so far. it is exactly what i meant but i did not realize that it being the same secret (which is technically just half implied by the grammar i think) is what breaks the dynamics.

if it's 1000 different secrets, then it kind of works but only if the guesser knows/assumes the distribution is uniform. (if it's the same secret then the guesser knows it's the same secret then that's the extreme, maybe a "degenerate" case, like having 1000 doors to one bank.)