158 MAGIC SQUARES [CH. VII 



the ninth order: it is written in a form which shows it as formed 

 by the superposition of two magic squares, one made with the 

 unit-digits and the other with the radix-digits. 



Trebly-Magic Squares. The construction of squares which 

 shall be magic for the original numbers, for their squares, and 

 for their cubes has also been studied. I know of no square of 

 this kind which is of a lower order than 128, and the construction 

 of a square of that order is not a " recreation." 



Other Magic Problems. Other problems, closely related 

 to magic squares, will suggest themselves; the following will 

 serve as specimens. 



Magic Card Square*. The first of these is the familiar 

 problem of placing the sixteen court cards (taken out of a pack) 

 in the form of a square so that no row, no column, and neither 

 of the diagonals shall contain more than one card of each suit 

 and one card of each rank. The solution presents no difficulty, 

 and is indicated in figure xxv below. There are 72 funda- 

 mental solutions, each of which by reflexions and reversals 

 produces 7 others. 



Euler's Officers Problem^. A similar problem, proposed by 

 Euler in 1779, consists in arranging, if it be possible, thirty-six 

 officers taken from six regiments — the officers being in six 

 groups, each consisting of six officers of equal rank, one drawn 

 from each regiment ; say officers of rank, a, b, c, d, e, f, drawn 

 from the 1st, 2nd, 3rd, 4th, 5th, and 6th regiments — in a solid 

 square formation of six by six, so that each row and each file 

 shall contain one and only one officer of each rank and one and 

 only one officer from each regiment. The problem is insoluble. 



Extension of Euler's Problem. More generally we may 

 investigate the arrangement on a chess-board, containing n. 2 



* Ozanam, 1723 edition, vol. iv, p. 434. 



t Kuler's Commentationes Arithmetical, Petrograd, 1849, vol. n, pp. 302 — 

 361. See also a paper by G. Tarry in the Comptes rendus of the French Associa- 

 tion for the Advancement of Science, Paris, 1900, vol. n, pp. 170 — 203 ; and 

 various notes in L'Intermediaire des MaMmaticiens, Paris, vol. in, 1896, 

 pp. 17, 90; vol. v, 1898, pp. 83, 176, 252; vol. vi, 1899, p. 251; vol. vn, 1900, 

 pp. 14, 311. 



