March 2, 1891.] 



KNOWLEDGE 



47 



The above types are distributed amongst the 880 

 squares in the following proportions : — 



880 



There is a close connection between some of the types, 

 especially between those of which there is the same 

 number of varieties. A simple transposition of the 

 figures will in many cases convert one into another. As 

 an instance of this, types G, I, J, and L are thus recipro- 

 cally convertible. If we take a G square and transpose 

 the two middle rows, and then the two middle columns, 

 it is turned into a square of the I type ; and if this I 

 square has its numbers diagonally adjacent to each other 

 (like those connected in diagram D) transposed, we get a 

 square of type J. If, further, this J square has its middle 

 rows and columns transposed, a square of type L is formed ; 

 and, once more altering the result like diagram D, we finally 

 get the same square of G with which we started, but 

 turned upside down. 



One way of estimating the number of squares of a 

 given type is to decompose the numbers into two skeleton 

 squares, each containing four numbers four times repeated. 

 By way of example, let us decompose a square of type C, 

 being the most numerous, putting the numbers 1, 2, 3, 4 

 into one and 0, 4, 8, 12 into the other skeleton square ; 

 thus : — 



(1) IS a square of type C; and (2) and (3) are two 

 skeleton squares which on being added together produce 



Now, on examining the structure of the squares (2) and 

 (3), we see that their own complementary pairs are 

 arranged in accordance with the type of (1), that is C. 

 In (2) the sum of each pair is 5 ; in (3) the sum is 12. 

 On further examination of the squares, wo may observe 

 that the numbers may be transposed in various ways so 

 as to produce difl'erent results, still belonging to the same 

 type. Thus, in square (2) each 2 may be written 3, and 

 ricf rersii, and each 4 may be written 1, and (■(<■(' rei\t,L 

 Tliis will give in combination with square (3) three addi- 

 tional squares of the same typo, or four in all. 



Furthermore, the numbers in scpiare (3) may be trans- 

 posed in similar manner, 4 for 8 and for 12, and ricr 

 rt'isii, giving another 4 varieties, which, combined with 

 the 4 varieties of the other skeleton square, give us 4x4, 

 or 10 squares of the type. Again, the arrangement of the 

 two skeleton squares may bo reversed ; we may write 

 down tlic 1, 2, 3, 4 in the way the 0, 4, 8, 12 are written 

 down above, and rice rcisd. 'L'liis doubles the number of 

 squares producible, giving us 10 x 2, or 82. Once more, a 

 partial transposition may be made in the numbers of 

 square (2). For example, the 2 and 3 may bo transposed 



in the top and bottom rows, whilst those in the middle 

 rows are undisturbed. 



This partial transposition may be performed on what- 

 ever numbers occupy the middle cells of the top and 

 bottom rows. This again doubles the number, bringing 

 it up to 32x2, or 64. 



Nor is this by any means all. There are two other 

 ways of decomposing these squares. Instead of putting 

 1, 2, 3, 4 into one, and 0, 4, 8, 12 into the other skeleton 

 square, we may put 1, 2, 5, 6 into one, and 0, 2, 8, 10 

 into the other ; or 1, 3, 5, 7 into one and 0, 1, 8, 9 into 

 the other. Thus 



The second set of series in (4) and (5), and the third 

 set in (7) and (8), have just the same arrangement re- 

 spectively as the numbers in (2) and (3). The resulting 

 squares (0) and (9), it may be noticed, are different from 

 (1) and different from each other. Using aU these three 

 sets of series therefore trebles the number of squares pi'e- 

 viously arrived at — gi\dng us 64 x 3, or 192. These 192 

 squares, however, do not exhaust the type C. I am in- 

 debted to Mr. James Cram, the author of an ingenious 

 little book on magic squares, for five other squares of this 

 type, each of which may be transposed in all the ways 

 above described excepting two, thus producing 16 varieties 

 of each instead of 04. This gives us 5 x 16, or 80 

 additional squares — making up the 272. 1 give below the 

 analysis of one of these 80, as it is very peculiar : 



1 12 10 114 4 4 8 12 



15 11 (! 2 3 3 2 2 12 8 4 



14 8 9 3 2 4 13 12 4 8 



4 10 7 13 4 2 3 1 8 4 12 



(10) (11) (12) 



The skeleton squares (11) and (12), it will be observed, 

 both sum wrongly in their diagonals. Nevertheless, on 

 combination, the resulting square is found to be correct ; 

 the errors having an opposite character, and neutralising 

 one another. 



The decomposition of squares is easily effected by aid of 

 little tables like the following : — 



Find the number you wish to decompose in the table 

 belonging to the series, and in a line with it will be found 

 at the top the number for one of the skeleton squares, and 

 at the left-hand side that for the other. 



A PERPETUAL CALENDAR. 

 Mr. C. L. Prinok, of Crowborough, has sent us a very 

 simple perpetual calendar devised by him a few years ago, 

 which avoids the necessity of committing to memory the 

 rather complicated rules given in Mr. R. W. D. Christie's 

 letter in our last number. 



The accompanying block may bo cut out and mounted 

 oil two pieces of cardboard. The inner circle of Domi- 



