124 



CHESS-BOARD RECREATIONS 



[CH. VI 



mathematical analysis was made by Euler* in 1759: it was 

 due to a suggestion made by L. Bertrand of Geneva, who sub- 

 sequently (in 1778) issued an account of it. This method is 

 applicable to boards of any shape and size, but in general 

 the solutions to which it leads are not symmetrical and their 

 mutual connexion is not apparent. 



Euler commenced by moving the knight at random over 

 the board until it has no move open to it. With care this 

 will leave only a few cells not traversed : denote them by 



a, b, His method consists in establishing certain rules 



by which these vacant cells can be interpolated into various 

 parts of the circuit, and by which the circuit can be made 

 re-entrant. 



The following example, mentioned by Legendre as one of 

 exceptional difficulty, illustrates the method. Suppose that 



Figure i. Figure ii. 



Example of Euler't Method. 



we have formed the route given in figure i above; namely, 1, 

 2, 3, . . . , 59, 60 ; and that there are four cells left untraversed, 

 namely, a, b, c, d. 



We begin by making the path 1 to 60 re-entrant. The 

 cell 1 commands a cell p, where p is 32, 52, or 2. The cell 60 

 commands a cell q, where q is 29, 59, or 51. Then, if any of 

 these values of p and q differ by unity, we can make the route 



* Hgmoires de Berlin for 1759, Berlin, 1766, pp. 310—337; or Commentationes 

 Arithmeticae Gollectae, Petrograd, 1849, vol. i, pp. 337 — 355. 



