NOTE ON MAGIC SQUARES. 
[From the Messenger of Mathematics, vol. vi. (1877), p. 168.] 
In a magic square of any odd order, formed according to the ordinary process, 
there is a tolerably simple analytical expression for the number which occupies any 
given compartment; thus taking the square of 21, let the dexter diagonals (N.W. to S.E.) 
commencing from the N.E. corner compartment, be numbered 1, 2, 3,.., 20, 21, 20', 
19',.., 2', 1', the diagonals of course containing these numbers of compartments respect 
ively ; and in any diagonal let the compartments reckoning from the top line be 
numbered 1, 2, 3,.., respectively; then if D 0j(/) (or D' e ^ as the case may be) denotes 
the number in the compartment 0 of the diagonal 0 or 0', we have 
D 
D\ 
20+1, <f> — 
20 ,</> — 
20+1, <£> 
D\ 
20 ,<#> 
200+ 10 + 0, 
200 + 231 + 0(- 21), 
- 220 + 430 + 0, 
- 220 + 231 + 0 (- 21), 
where in the second and fourth expressions the term — 21 is to be retained only if