CH VII] 



MAGIC SQUARES 



151 



any two columns, and the two pieces be interchanged, the new 

 square so formed will be also pandiagonally magic. Hence it is 

 obvious that by one vertical and one horizontal transposition of 

 this kind any number can be made to occupy any specified cell. 



Pandiagonal magic squares of an odd order can be con- 

 structed by a rule somewhat analogous to that given by De la 

 Loubere, and described above. I proceed to give an outline of 

 the method. 



If we write the numbers in the scale of notation whose 

 radix is n, with the understanding that the unit-digits run from 

 1 to n, it is evident, as in the corresponding explanation of why 

 De la Loubere's rule gives a magic square, that all we have to 

 do is to ensure that each row, column, and diagonal (whether 

 broken or not) shall contain one and only one of each of the 

 unit-digits, as also one and only one of each of the radix-digits. 



Figure xvi. Figure xvii. Pandiagonal Square, n = 5. 



This is seen to be the case in the square of the fifth order 

 delineated above in figures xvi and xvii. 



Let us suppose that we write the numbers consecutively, 

 and proceed from cell to cell by steps, using the term step 

 (a, b) to denote going a cells to the right and b cells up. Thus 

 a step (a, b) will take us from any cell to the ath column to the 

 right of it, to the 6th row above it, to the (6 + a)th diagonal 

 above it sloping down to the right, and to the (b — a)th diagonal 

 above it sloping down to the left. In all cases we have the 

 convention, as in De la Loubere's rule, that the movements 

 along lines are taken cyclically; thus a step n + a is equivalent 

 to a step a. Of course, also, if a means going a cells to the 

 right, then - a will mean going a cells to the left ; thus if the 



