62 



THE MAGIC SQUARE. 



and the numbers then taken off in layers of 16 from the top down- 

 wards. We obtain thus 4 squares of 16 cells each, which together 

 make up the magic cube ; as the following diagrams will show : 



First Layer 

 from the Top. 



Second Layer 

 from the Top. 



Third Layer 

 from the Top. 



Fourth Layer 

 from the Top. 



The same sum 130 here comes out not less than 52 times ; viz. 

 in the first place from the 16 rows from left to right, secondly from 

 the 1 6 rows from the front to the back, thirdly from the 16 rows 

 counting from the top to the bottom, and lastly from the 4 rows 

 which join each two opposite corners of the cube, namely from the 

 rows: i, 43, 22, 64; 49, 27, 38, 16 ; 13, 39, 26, 52; 61, 23, 42, 4. 



For a cube with 5 compartments in each edge the arrangement 

 of the figures can so be made that all the 75 rows parallel to any and 

 every edge, all the 30 rows lying in any diagonal of a square, and 

 all the 4 rows forming any principal diagonal shall have one and the 

 same summation, 315. 



Just as the magic squares of an odd number of cells could be 

 iormed with the aid of two auxiliary squares, so also odd-numbered 

 magic cubes can be constructed with the help of three auxiliary cubes. 



First Layer from Top. Second Layer from Top. Third Layer from Top. 



Fourth Layer from Top. 



Lowest Laver. 



