146 MAGIC SQUARES [CH. VII 



cise form. One rule in the case of an odd square, where 

 n=2m+ 1, may be put as follows. First, write in the n cells 

 in the top row from left to right the numbers 2, 1, m + 1, ..., 

 2m -1, 3m + 1, 3m + 2, ..., 4m. Next, write in the n cells in 

 the left-hand column from top to bottom the numbers 2, 3, . . ., m, 

 2m, 1m + 1, . . ., 3m, 4m 2 + 2. Fill up the remaining border cells 

 with the proper complementary numbers. Next take the m 

 numbers in the upper row immediately to the right of the left- 

 hand top corner cell, and the m numbers in the left-hand 

 column immediately above the left-hand bottom corner cell. 

 Interchange the numbers in these 2m cells with the numbers 

 in their complementary cells. The resulting square is magic. 

 It may interest my readers to see if they can evolve a similar 

 simple rule for the formation of bordered even squares. 



Alternative methods of constructing simple magic squares 

 are constantly being expounded, so the subject is not exhausted, 

 but there is no occasion to go here into further details. 



Number of Squares of a Given Order. One unsolved 

 problem in the theory is the determination of the number of 

 magic squares of the fifth (or any higher) order. There is, in 

 effect, only one magic square of the third order, though by 

 reflexions and rotations it can be presented in 8 forms. There 

 are 880 magic squares of the fourth order, but by reflexions and 

 rotations these can be presented in 7040 forms. The problem 

 of the number of magic squares of an order higher than five 

 is unsolved. From the square given above in figure iii on 

 page 139 formed by De la Loubere's method, we can get 720 

 other distinct squares, for we can permute the symbols 1, 2, 3, 

 4, 5, in 5! ways, and combine with any of these squares any of 

 the 4! squares obtained by permuting the symbols 0, 5, 15, 20. 

 We thus obtain 2880 magic squares of the fifth order, though 

 only 720 of them are distinct. Bachet gave a somewhat similar 

 construction in which he began by placing 1 in the cell im- 

 mediately above the middle cell, and wrote the consecutive 

 numbers in a line sloping downwards : his method gives another 

 720 distinct magic squares of the fifth order. There are however 

 numerous other rules for constructing odd magic squares, and 



