126 



CHESS-BOARD RECREATIONS 



[CH. VI 



Hence we must replace the numbers 1, 2, ..., 13 by 13, 12, ..., 

 1, and we obtain a re-entrant route covering the whole board, 

 which is represented in the second of the diagrams given above. 

 Euler showed how seven other re-entrant routes can be deduced 

 from any given re-entrant route. 



It is not difficult to apply the method so as to form a route 

 which begins on one given cell and ends on any other given 

 cell. 



Euler next investigated how his method could be modified 

 so as to allow of the imposition of additional restrictions. 



An interesting example of this kind is where the first 32 

 moves are confined to one-half of the board. One solution 

 of this is delineated below. The order of the first 32 moves 



Euler 1 ! Half-board Solution. 



Boget's Half-board Solution. 



can be determined by Euler's method. It is obvious that, if 

 to the number of each such move we add 32, we shall have 

 a corresponding set of moves from 33 to 64 which would cover 

 the other half of the board ; but in general the cell numbered 

 33 will not be a knight's move from that numbered 32, nor will 

 64 be a knight's move from 1. 



Euler however proceeded to show how the first 32 moves 

 might be determined so that, if the half of the board con- 

 taining the corresponding moves from 33 to 64 was twisted 

 through two right angles, the two routes would become united 

 and re-entrant. If a; and y are the numbers of a cell reckoned 

 from two consecutive sides of the board, we may call the cell 



