246 Kansas Academy of Science. 



placed in the alternate diagonal cells from each of them re- 

 spectively. 



3. — (a) Place 5 in either one of the two remaining cells be- 

 side 16, and {h) its mate, 6, in the same position relatively to 

 5 as 2 is to 1; that is to say, in an alternate line and adjacent 

 column, (c and d) Their complements, 12 and 11, are to be 

 placed, as before, in the alternate diagonal cells from each of 

 them. 



4. — (a) Place 9 in the only remaining cell next to 16, and 

 (6) its mate, 10, in the same relative position to it as 2 is to 1. 

 (c and d) Their complements, 8 and 7, are to be placed as be- 

 fore in the alternate cells diagonally from them. 



The numbers 2, 3, 5 and 9 are always to be laid around 16. 

 The reason for this is that these four places are just a "knight- 

 step" or paladin step from 1 ; all other cells in the square are in 

 the same line with 1, either vertically, horizontally, or di- 

 agonally ; 16 is at the intersection of the two diagonals. The 

 four numbers mentioned must occupy those four cells and no 

 other. They may be laid in any order. The number 9 may be 

 placed first if preferred, and may be placed in any of the four 

 vacant cells next to 16. It is immaterial what order these 

 four numbers are placed in; it is material where their mates 

 are placed. 



Once an antecedent number is placed in a cell there is only 

 one place for its mate, according to the scheme, and one place 

 for each of their complements. There are really four places 

 in either of which a consequent of a couplet may be placed; 

 but there are four schemes; and whatever scheme is adopted 

 for the first couplet must be followed for all the rest, in order 

 to preserve the unity and harmony of the square. 



So the placing of 2 immediately predetermines the position 

 of seven other numbers; for the position of every consequent 

 of a couplet must bear the same relation to its antecedent that 

 2 bears to 1. Mates are always a paladin step apart, that is, 

 two cells in one direction and one cell at right angles to the 

 two ; for while they may appear otherwise, the one is a paladin 

 step from the other across the margin of the square, as though 

 the other were in its proper position in an adjoining square. 

 The real position of the consequents, however, depends upon 

 the positions occupied by 3 and 5, in the placing of each of 

 which there is some latitude. 



Similarly, the placing of 3 predetermines the position of 



