Most-perfect squares David S. Br\'{e}e\ and Dame Kathleen Ollerenshaw We have developed a method for constructing and enumerating all squares of any size in an interesting class of magic squares, which we call 'most-perfect.' This is the first time, in the thousand years during which magic squares have challenged mathematicians, that a method of construction, let alone enumeration, has been found for a whole class of magic square. In most-perfect squares integers come in complementary pairs along the diagonals and any 2x2 block of four adjacent integers add to the same sum. These properties ensure that most-perfect squares are pandiagonal magic squares. There is a remarkably large number of these squares. In this talk we will: - give a brief state-of-the-art in the construction of pandiagonal magic squares before our work - demonstrate the method of constructing most-perfect squares by transformation from 'reversible squares' - show how all reversible squares can be constructed - indicate why there is a one-to-one correspondence between most-perfect and reversible squares - show how reversible squares can always be enumerated as the sum of a series of products of binomial coefficients - show the closed form solution for the sum for n a power of 2 - give a simplified (but not-closed form) solution for the sum when n=2^r p^s (p any prime>2; r, s integers >0) - open up the possibilities for the application of most-perfect squares The talk is based on the book: Most-perfect pandiagonal magic squares: their construction and enumeration. IMA, Southend-on-Sea, 1998. ISBN 0-905091-06-X Available in Blackwell's bookshop.