Magic squares of subtraction of Adam Adamandy Kochański

Abstract

The problem of the construction of magic squares occupied many mathematicians of the 17th century. The Polish Jesuit and polymath Adam Adamandy Kochański studied this subject too, and in 1686 he published a paper in Acta Eruditorum titled "Considerationes quaedam circa Quadrata et Cubos Magicos". In that paper he proposed a novel type of magic square, where in every row, column and diagonal, if the entries are sorted in decreasing order, the difference between the sum of entries with odd indices and those with even indices is constant. He called them \emph{quadrata subtractionis}, meaning squares of subtraction. He gave examples of such squares of orders 4 and 5, and challenged readers to produce an example of square of order 6. We discuss the likely method which he used to produce squares of order 5, and show that it can be generalized to arbitrary odd orders. We also show how to construct doubly-even squares. At the end, we show an example of a square of order 6, sought by Kochański, and discuss the enumeration of squares of subtraction.