On certain General Limitations affecting Hyper-magic Squares
Samuel Kelton Jr. Roberts · Proceedings of the London Mathematical Society · 1892
1.This paper does not aim tit making any addition to tho known •ways of constructing magio squares.*Hyper-magic squares, as I regard them, include thoso called by tho late M. 1. 3. Luousf " carrcs diaboliques," and also treated of by Rev. A. H. Frost, under the designation "nasik squares."!The special form is of ancient origin.The second method given in the fragment by Moschopulus (probably of the fourteenth century) is a general one lor forming such squares, and they have been discussed by various modern authors.My object is to show some limitations to which they are subject when tho elements are positive or negative integers.Incidentally it will appear that hyper-mngio squares of oddly even orders cannot be formed of series of consecutive natural numbers.§ There is some reason to believe that much ingenuity has * Notwithstanding this rcirmrk, it has boon imagined that I contemplated tho actual construction of hyper-magic squares having consecutive natural nuiuhcrH as elements.Ho lar is thin from being tho case, 1 huvo not, unless inadvertently, shown that such squares exist.It was not necessary, sinco my conclusions aro of a negative kind.t Tho subject has been brought into connexion with tho " Geometry of Tissues," by M. LuctiR, and others (1'rincipii fondiunmlali ilella Gcomclria dci Tcssuli, per Edoardo Lucas, Torino, 1880 ; HOO also Recreations Malhvtnaliques, par M. K. Lucas, Introduction, t. i., p. xviu.).$ I do not say that hypor-magie squares includo nasik squares, but that thoy include " cam's diaboliquos," wiiic.li,I take it, aro hypor-mugio squares mado up of natural numbers from I to n' J (v. } 2).The lirst definition of nasik squares {Quarterly Journal of Mathematics, vn., pp.!).' {, 91) apparently makes " carres diaboliipios " cooxtenHivo with them.A later definition {Quay.Jour., xv., p. M) IH in the following terms: " A square containing n cells on each side, in which aro placed the natural numbers from 1 to » 2 , in such an order that tho constant sum J« (»»' -+ 1) is obtained by adding the numbers on n of tho colls, those n cells lying in a variety of difusront directions, and thoir relative position in each direction being defined by simple laws."I should not presume to limit tho comprehensiveness of thin definition.§ "With regard to this, I havo been referred to tho following passago in Itov. A. If.Frost's paper {Quarterly Journal nf Mnlheinaliim, xv., p. 1!)) :-" Nawik Squares of the form 2 (2M + 1) cannot be filled with eonsecutivo natural numbers from 1 to 4 (2» + I) 2 either by this or the process adopted in tho previous paper; for it will bo found that, as in the squares of the form 4M, we have to givo {c.ff.y tho case of C 3 ) p\,p%, .../>« such values that the Hum of 3 equals tho sum of tho * Sco additional noln nt tho end of this paper.* 'J'ho definition of " carrtfs diaboliquos" given by M. Lucas (Itecriations Math., Jntiofliiction, t. i., p. xvn.) in founded on this property.