THE MAGIC SQUARE. 



47 



of the two numbers in any two corresponding cells each combina- 

 tion shall come out once and only once ; and further that in each 

 horizontal, vertical, and diagonal row in each auxiliary square each 

 number shall once appear. Then the required sum of 65 must 

 necessarily result in every case, because the numbers from i to 5 

 added together make 15, and the numbers o, 5, 10, 15, 20 make 50. 

 We effect the required method of inscription by imagining the 

 numbers i, 2, 3, 4, 5 (or o, 5, 10, 15, 20) arranged in cyclical suc- 

 cession, that is i immediately following upon 5, and, starting from 

 any number whatsoever, by skipping each time either none or one 

 or two or three etc. figures. Cycles are thus obtained of the first, the 

 second, the third etc. orders ; for example 3 4 5 i 2 is a cycle of the 

 first order, 241 3 5 is a cycle of the second order, i 5 4 3 2 is a 

 cycle of the fourth order, etc. The only thing then to be looked out 

 for in the two auxiliary squares is, that the same "cycle" order be 

 horizontally preserved in all the rows, that the same also happens for 

 the vertical rows, but that the cycle order in the horizontal and ver- 

 tical rows is different. Finally we have only additionally tf> take 

 care that to the same numbers of the one auxiliary square not like 

 numbers but different numbers correspond in the other auxiliary 

 square, that is lie in similarly situated cells. The following auxiliary 

 squares are, for example, thus possible : 



and 



Fig. 8. 



Fig. 9. 



Adding in pairs the numbers which occupy similarly situated 

 ceils, we obtain the following correct magic square : 



Fig. 10. 



