THE MAGIC SQUARE. 6l 



and along the six that are constructed at right angles to its sides ; 

 this sum, for the first hexagon from within, is in, for the second 

 185, and for the third 259. 



i 5 6 70 60 59 58 

 63 8 



62 19 53 46 22 45 9 



61 20 24 64 



2 48 3i 42 38 49 57 



3 47 39 4 44 56 



67 51 4 1 37 33 23 7 



66 50 34 35 54 n 



65 25 36 32 43 26 12 



10 30 27 13 



17 29 21 28 52 55 72 



18 71 



16 69 68 4 14 15 73 

 Fig. 34- 



X. 



MAGIC CUBES. 



Several inquirers, particularly Kochansky (1686), Sauveur 

 (1710), Hugel (1859), and Scheffler (1882), have extended the prin- 

 ciple of the magic squares of the plane to three-dimensioned space. 

 Imagine a cube divided by planes parallel to its sides and equidistant 

 from one another, into cubical compartments. The problem is then, 

 so to insert in these compartments the successive natural numbers 

 that every row from the right to the left, every row from the front 

 to the back, every row from the top to the bottom, every diagonal 

 of a square, and every principal diagonal passing through the centre 

 of the cube shall contain numbers whose sum is always the same. 

 For 3 times 3 times 3 compartments, a magic cube of this descrip- 

 tion is not constructible. For 4 times, 4 times 4 compartments a 

 cube is constructible such that any row parallel to an edge of the 

 cube and every principal diagonal give the sum of 130. To obtain 

 a magic cube of 64 compartments, imagine the numbers which be- 

 long in the compartments written on the upper surface of the same 



