THE MAGIC SQUARE. 



six times the numbers from i to 6 that each number shall appear once 

 and only once in each horizontal, vertical, and diagonal row ; tor 

 example, in the following manner : 



Fig. 18. 



But if we attempt so to insert, in a like manner, the other set of 

 numbers o, 6, 12, 18, 24, 30 in a second auxiliary square, that each 

 number of the first auxiliary square shall stand once and once only 

 in a corresponding cell with each number of the second square, all 

 the attempts we may make to fulfil coincidently the last named con- 

 dition will result in failure. It is therefore necessary to select aux- 

 iliary squares like the two given above. It is noteworthy, that the 

 fulfilment of the second condition is impossible only in the case of 

 the square of 6, but that in the case of the square of 4 or of the 

 squaie of 8, for example, two auxiliary squares, such as the method 

 of De la Hire requires, are possible. Thus, taking the square of 4 

 we get 



^ 3 *- ^ *"" ~~ "T 



Fig. 19. 



Fig. 20. 



The reader may form for himself the magic square which these 



give. 



The existence of these two auxiliary squares furnishes a key to 

 the solution of a pretty problem at cards. If we replace, namely, 

 the numbers i, 2, 3, 4 by the Ace, the King, the Queen, and the 

 Knave, and the numbers o, 4, 8, 12 by the four suits, clubs, spades, 

 hearts, and diamonds, we shall at once perceive that it is possible, 

 and must be so necessarily, quadratically to arrange in such a man- 

 ner the four Aces, the four Kings, the Four Queens, and the fou*. 



