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(Unanswered duplicate here, from 9 years ago)

Suppose we're trying to brute force a multiple-choice test. Each question $i$ has $A_i$ answers, and there is only one right answer for each question. We are allowed unlimited attempts, but after each attempt, we are only given the amount of questions we got right, but not which ones.

The question is: How many tests do we need to take, in the worst case scenario? Is there an algorithm that allows us to do this?

A simple calculation suggests that the lower bound is $\lceil log_{(q-1)}(P) \rceil$, where $q$ is the number of questions, and $P=\prod\limits_{n=1}^{q}A_n$. Unfortunately, the formula only works for $q\ge3$. Also, it generates a very low numbers, indicating it can be improved.

undefned
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    This case where each question has two answers, a true/false test, has been asked before: https://math.stackexchange.com/questions/1058194/ $\tag*{}$The case where all questions have the same number of options was asked earlier as well. The answer given applies to three option tests, and claims to generalize to any prime number of options: https://math.stackexchange.com/questions/4133092/a-confusing-test-with-n-questions-with-k-choices-that-can-only-be-guessed – Mike Earnest Oct 25 '24 at 22:36

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