60 



KANSAS ACADEMY OF SCIENCE. 



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Scheme VII ffigr. 41) is a variation of scheme III, and 

 differs from it as scheme V differs from scheme I. 



No. .51. 



1 8 

 Scheme VIII (fig. 42) differs from scheme IV, as scheme 7 2 



V differs from scheme I. (See square No. 51.) ^ | 



No. 52. 



In scheme IX (fig. 43) the lines of the trial arrange- 

 ment govern the outer opposite lines of the squares, in- 

 stead of alternate or adjacent lines, as in former schemes. 

 That is to say, the lines of the trial arrangement as they 

 stand are to be applied to the first and fourth lines; 

 their doubles, then, will fall in the outer lines of the 

 square to be formed, and the mates in the inner lines. 



No. 53. 



In scheme X (fig. 44), also, the trial arrangement ' — ' — 7 

 governs the outer opposite lines. (See square No. 53.) 8 2 3 5 



No. 54. 



Scheme XI — Knight's Tour. — This scheme (fig. 



45) is not harmonic: in a square of four it gives or- 



17 5 4 ^- . - ^„ dinary magic squares. Its defect is that it throws 



^235 = in ,,. i four even numbers into one diagonal and none into 



the other, thereby increasing that diagonal by 2 and 

 decreasing the other by the same amount. But in a 

 square of eight it does very well; because opposite quarters can be easily fitted 

 to balance. The scheme differs from scheme X in reversing or inverting (tech- 

 nically condiverting) the lower half. 



