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9 must occur admits the configuration admitting all paths Aij»k Aijk Arrange the integers array of Lemma carry principal paths column occur opposite coordinates Corollary cross ratio cube of order d.c. of order define Cg determined diabolic square exactly one cross f g h Hence Irregular n.d.s. key cubes latin squares lattice L(i,J;q least prime factor Let three x»s magic cubes MAGIC SQUARES main diagonal matrix mh-th column mod n mod p°m mod pa+1m normalized n.p.s. number of normalized occur at 3,5 occurs at 5,9 order 8m order for J£L order mn order n admits order q paths but rows paths into principal paths P(l,ax prime order quadratic residues R. J. WALKER ROSSER and R. J. rows and columns s-path s-rowed minors saqno second column occur second column opposite seqno sets of paths square admitting square of order Suppose THEOREM three x»s occur totally diabolic Transformations which carry zero