Self-Referential Tests

Abstract

We study self-referential multiple-choice tests with the question: \emph{How many correct answer choices are there?} The answer choices are positive integers. A value $a$ is called \emph{valid} if it occurs exactly $a$ times among answer choices. The \emph{cost} of a test is the sum of all answer choices, linking the problem to integer partitions. Using this framework, we define solvable and $k$-solvable tests and derive generating functions that enumerate them by cost, number of distinct valid values, and number of options. We also investigate extremal questions, including minimum costs and the maximum possible number of valid values. The paper was inspired by a puzzle from \emph{Mathematical Puzzles and Curiosities}.