Immensely Questionable Tests

I have finally become a queen — though only in a research paper. In Immensely Questionable Tests, I am Queen Tanya, and my ten PRIMES STEP student coauthors are my sages. The project began with a probability puzzle from Mathematical Puzzles and Curiosities, my book with Ivo David and Yogev Shpilman. Here I present an equivalent version, without probabilities, to match our project. Puzzle. How many correct answer choices are there? a) 1 b) 1 c) 2 d) 2 The intended answer is 2: exactly two choices say 2. So far, so good. Now erase one of the 1s. How many correct answer choices are among a) 1, b) 2, and c) 2? Now 1 appears once and 2 appears twice. Either value can be an answer. But calling all three choices correct would require the answer to be 3. What is going on here? We restrict the answer choices to positive integers. We call a value m valid if it occurs exactly m times among the choices. Now we can discuss valid answers without getting into paradoxes. Our first paper…