146 BULLETIN OF THE 



therefore the 1st line must be one of the diagonals, and one of the 

 other three lines the other diagonal. 



Making a -|- a' = b + b' &c.=37, and putting in the conditions 

 for a magic square we have : 



a4-bJ-c + d = 104, 99 or 94 

 e + f + g + h = 92, 89 or 86 

 k + 1 + m + n = 80, 79 or 78 

 e — h + k — n= 5, 10 or 15 

 a — d + 1 — m= 3, 6 or 9 

 b — c+f — g= 1, 2or 3 



Whence, by addition, 2 (a + b-feH-fH-k-fl) = 285, an odd 

 number, which is impossible. 



If we had constructed the square having 4 squares on each side 

 in a similar way, we should have obtained two solutions, but in one of 

 the solutions one of the numbers would have been used twice, and 

 another one not at all, and so, for that reason it would have been 

 excluded, therefore the one given is the only symmetrical solution ; 

 but there will be eight ways of arranging it, viz : by beginning at 

 each corner and filling to the right or left. 



The construction is very simple, for of the eight ways of doing 

 it we will commence one below the middle square and move diagon- 

 ally to the right, taking care when we get to the bottom to suppose 

 the whole square moved down and begin again at top ; when we go 

 over to the left, suppose the whole square moved over, &c,, begin- 

 ning again at the right, and, when one square is filled up, move 

 from there diagonally to the left, or, what amounts to the same 

 thing, move down two vertically from your last number, the square 

 will be entirely symmetrical ; the sum of any two figures similarly 

 placed with reference to the centre will be 122. There are many 

 more interesting features connected with it that I leave to the 

 reader's ingenuity to suggest. 



