150 MAGIC SQUARES [CH. VII 



the points of their intersection with the cross-joins be joined 

 cross- wise, these new cross-joins will intersect on the axis of the 

 original pencil or on lines perpendicular to it. The whole figure 

 will now give 8 s lines, arranged in 244 pencils each of 8 rays, 

 and will be the reciprocal of a magic cube of the 8th order. If 

 we reciprocate back again we obtain a representation in a plane 

 of a magic cube. 



Hyper-Magic Squares. With the exception of determining 

 the number of squares of a given order, we may fairly say that 

 the theory of the construction of magic squares, as defined above, 

 has been sufficiently worked out. Accordingly attention has of 

 late been chiefly directed to the construction of squares which, 

 in addition to being magic, satisfy other conditions. I term 

 such squares hyper-magic. Of hyper-magic squares, I will deal 

 only with the theory of Pan-Diagonal and of Symmetrical 

 Squares, though I will describe without going into details what 

 are meant by Doubly and Trebly Magic Squares. 



Pandiagonal Squares. One of the earliest additional con- 

 ditions to be suggested was that the square should be magic 

 along the broken diagonals as well as along the two ordinary 

 diagonals*. Such squares are called Pandiagonal. They are 

 also known as nasik, perfect, and diabolic squares. 



For instance, a magic pandiagonal square of the fourth 

 order is represented in figure ii on page 137. In it the sum 

 of the numbers in each row, column, and in the two diagonals 

 is 34, as also is the sum of the numbers in the six broken 

 diagonals formed by the numbers 15, 9, 2, 8, the numbers 

 10, 4, 7, 13, the numbers 3, 5, 14, 12, the numbers 6, 4, 11, 13, 

 the numbers 3, 9, 14, 8, and the numbers 10, 16, 7, 1. 



It follows from the definition that if a pandiagonal square 

 be cut into two pieces along a line between any two rows or 



* Squares of this type were mentioned by P. De la Hire, J. Sauveur, and Euler. 

 Attention was again called to them by A. H. Frost in the Quarterly Journal of 

 Mathematics, London, 1878, vol. xv, pp. 34 — 49, and subsequently their properties 

 have been discussed by several writers. Besides Frost's papers I have made 

 considerable use of a paper by E. McClintock in the American Journal of Mathe- 

 matics, vol. xix, 1897, pp. 99—120. 



