The question of what role free schools should perform in the Swedish educational system has been a contested subject between three ideological, theoretical, political and policy tenets. The first, "contribution to pedagogical diversity in a controlled school market" reflects a traditional social democratic view. The second, "contribution to a better education on a competitive school market" reflects a neo-liberal approach. The third tenet, "contribution to the maintenance of groups' and individuals' cultural and religious identity" reflects a multicultural view insisting on the thesis that a family's cultural and religious identity should be a steering motive for the school choice. The aim of this article is to take a closer look at what the three tenets that constitute the "riddle" contain--including their claims, responses to critics, arguments and empirical evidence--and to discuss some of their practical impacts on the shaping of educational policy. (Contains 6 notes.)