Agnes Scott College

Dame Kathleen Ollerenshaw and Sir Hermann Bondi, F.R.S.

Magic Squares of Order Four
Philosophical Transactions of the Royal Society of London, Vol. 306 (October 15, 1982), 443-532


A brief history of work on the 4 x 4 magic square is presented, with particular reference to Frénicle's achievement over 300 years ago of establishing 880 as the number of essentially different squares by using the method of exhaustion (not convincingly repeated except by computer in 197). He also established several central theorems. Our paper confirms the number 880 by a wholly new method of "Frénicle quads" and "part sums", which leads to the classification of all solutions into, initially, six "genera" one of which has no members and thence to the enumeration of all possible solutions by analytical methods only. The working leads also to the first analytical proof independent of solutions that 12 and only 12 patterns formed by linking "complementary" numbers within a square are necessary and sufficient to describe all solutions – a fact which has been known since 1908, but not hitherto proved. A second method of construction and partial proof, greatly shortened by what has gone before, is also described. This yields a highly symmetrical list of the 880 magic squares. Together the two methods combine to explain many of the special characteristics and otherwise mysterious properties of these fascinating squares. The complete symmetrical list of squares ends the paper.

  1. The Magic Square of Order Four and Its History
  2. The Groundwork
    1. Frénicle's square
    2. Frénicle's rules
    3. Quads
    4. Quadsets
    5. Part sums
    6. Parities within quads
    7. Quads with a common part sum
    8. Transformations U, S, US
    9. Complements
    10. Reversals
    11. Quadsets formed by two pairs of mutually-complementary quads
  3. The Multiplicities of Frénicle Quadsets by Genus
    1. Constructing solutions – the cross
    2. The Frénicle-quadset genera
    3. Self-complementary quadsets
    4. Non-self-complementary quadsets
  4. Enumeration of the Frénicle Quadsets
    1. Populations
    2. The populations for genera Π, X, Δ, Ω – the arrays
    3. The populations for the genera Θ, Φ
    4. Summary of enumerations
  5. The Alternative "Matrix Method"
  6. The Magic Card and the Boss Problems
  7. Summary of Significant or Curious Facts
  8. Appendices