Mathematical recreations and essays . s A,B, ...^H; 8 pencils, with verticesA,...,H; one pencil with its vertex at 0; and one pencil withits vertex on the axis of the last-named pencil. The sum of the numbers in each of the 18 lines in figure xiiis the same. To make figure xv correspond to this we mustnumber the lines in the pencil A from left to right, 1, 9, ..., 57, * Communicated to me by R. Strachey. t See Magic Reciprocals by G. Frankenstein, Cincinnati, 1875. CH. VIl] MAGIC SQUARES 149 following the order of the numbers in the first column of thesquare: the lines in pencil B must be numb
Mathematical recreations and essays . s A,B, ...^H; 8 pencils, with verticesA,...,H; one pencil with its vertex at 0; and one pencil withits vertex on the axis of the last-named pencil. The sum of the numbers in each of the 18 lines in figure xiiis the same. To make figure xv correspond to this we mustnumber the lines in the pencil A from left to right, 1, 9, ..., 57, * Communicated to me by R. Strachey. t See Magic Reciprocals by G. Frankenstein, Cincinnati, 1875. CH. VIl] MAGIC SQUARES 149 following the order of the numbers in the first column of thesquare: the lines in pencil B must be numbered similarly tocorrespond to the numbers in the second column of the square,and so on. To prevent confusion in the figure I have not insertedthe numbers, but it will be seen that the method of constructionensures that the sum of the 8 numbers which designate thelines in each of these 18 pencils is the same. We can proceed a step further, if the resulting figure is cutby two other parallel lines perpendicular to the axis, and if all. B C D E ^ G Figure xv. A Magic Pencil, w=8. 150 MAGIC SQUARES [CH. VII the points of their intersection with the cross-joins be joinedcross-wise, these new cross-joins will intersect on the axis of theoriginal pencil or on lines perpendicular to it. The whole figurewill now give 8^ lines, arranged in 244 pencils each of 8 rays,and will be the reciprocal of a magic cube of the 8th order. Ifwe reciprocate back again we obtain a representation in a planeof a magic cube. Hyper-Magic Squares. With the exception of determiningthe number of squares of a given order, we may fairly say thatthe theory of the construction of magic squares, as defined above,has been sufficiently worked out. Accordingly attention has oflate been chiefly directed to the construction of squares which,in addition to being magic, satisfy other conditions. I termsuch squares hyper-magic. Of hyper-magic squares, I will dealonly with the theory of Pan-Diagonal and of SymmetricalSquar
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