132 



CHESS-BOARD RECREATIONS 



[CH. VI 



Roget's method can be also applied to two half-boards, as 

 indicated in the figure given above on page 126. 



The method which Jaenisch gives as the most fundamental 

 is not very different from that of Roget. It leads to eight 

 forms, similar to that in the diagram printed below, in which 

 the sum of the numbers in every column and every row is 260 ; 

 but although symmetrical it is not in my opinion so easy to 

 reproduce as that given by Roget. Other solutions, notably 

 those by Moon and by Wenzelides, were given in former 

 editions of this work The two re-entrant routes printed below, 

 each covering 32 cells, and together covering the board, are 

 remarkable as constituting a magic square* 



Jaenisch's Solution. 



Two Half Board Solutions. 



It is as yet impossible to say how many solutions of the 

 problem exist. Legendref mentioned the question, but Minding J 

 was the earliest writer to attempt to answer it. More recent 

 investigations have shown that on the one hand the number of 

 possible routes is less§ than the number of combinations of 168 

 things taken 63 at a time, and on the other hand is greater 

 than 31,054144 — since this latter number is the number of 

 re-entrant paths of a particular type||. 



* See A. Billy, Le Problime du Cavalier des Echecs, Troyes, 1905. 

 + TMorie des Nombres, Paris, 2nd edition, 1830, vol. n, p. 165. 

 J Cambridge and Dublin Mathematical Journal, 1852, vol. vn, pp. 147 — 156; 

 and Crelle's Journal, 1853, vol. xliv, pp. 73 — 82. 

 § Jaenisch, vol. n, p. 268. 

 || Bulletin de la Sociite Mathimatique de France, 1881, vol. ix, pp. 1 — 17. 



L'^} UO. { - ' 



