I think you're assuming that you can count the jar between pulls. (That was my initial reading also.) In that case, you guess your first number, count the jar,

I think you're assuming that you can count the jar between pulls. (That was my initial reading also.) In that case, you guess your first number, count the jar, and then work backwards for all subsequent pulls. But on rereading, the info to prove isn't "I know how many beans were here to start", it's "I'm capable of counting this jar of beans by looking at it". So if you guess right on the first trial, you still won't know the beans-remaining count, and will have to guess on the next trial too. This is made super confusing by the claim that it's a ZKP of the initial count. That's narrowly true, but the real question at hand is whether you can count the beans.