It is not often that one can introduce primary school students to a problem at the forefront of mathematics research, and have any expectation of understanding; but with magic squares, one can do exactly that. Magic squares are an ideal tool for the effective illustration of many mathematical concepts. This paper assumes little prior knowledge on the part of the student except for addition and multiplication, reflection and rotation; but, using questions and exercises throughout, finishes by posing a problem tough enough to test postgraduate mathematics students, to which no-one has yet managed to find a solution. In this article, the author invites students and their teachers to engage with the mathematics of magic squares. (Contains 13 figures.)