128 CHESS-BOARD RECREATIONS [CH. VI 



4/4, 5/6, 7/5, 8/7, 6/8, 7/6, 8/8, 6/7, 8/6, 7/8, 5/7, 6/5, 8/4, 

 7/2, 5/1, 6/3. 



The path is re-entrant but unsymmetrical. Had he trans- 

 ferred the first three fractions to the end of this series he 

 would have obtained two symmetrical circuits of thirty-two 

 moves joined unsymmetrically, and might have been enabled 

 to advance further in the problem. Vandermonde also con- 

 sidered the case of a route in a cube. 



In 1773 Collini* proposed the exclusive use of symmetrical 

 routes arranged without reference to the initial cell, but con- 

 nected in such a manner as to permit of our starting from 

 it. This is the foundation of the modern manner of attacking 

 the problem. The method was re-invented in 1825 by Prattf, 

 and in 1840 by Roget, and has been subsequently employed 

 by various writers. Neither Collini nor Pratt showed skill in 

 using this method. The rule given by Roget is described later. 



One of the most ingenious of the solutions of the knight's 

 path is that given in 1823 by Warnsdorff J. His rule is that 

 the knight must be always moved to one of the cells from 

 which it will command the fewest squares not already traversed. 

 The solution is not symmetrical and not re-entrant ; moreover 

 it is difficult to trace practically. The rule has not been 

 proved to be true, but no exception to it is known : apparently 

 it applies also to all rectangular boards which can be covered 

 completely by a knight. It is somewhat curious that in most 

 cases a single false step, except in the last three or four moves, 

 will not affect the result. 



Warnsdorff added that when, by the rule, two or more cells 

 are open to the knight, it may be moved to either or any of 

 them indifferently. This is not so, and with great ingenuity 

 two or three cases of failure have been constructed, but it 

 would require exceptionally bad luck to happen accidentally 

 on such a route. 



* Solution du ProbUme du Cavalier au Jeu des Echecs, Mannheim, 1778, 

 t Studies of Chess, sixth edition, London, 1825. 



% H. C. Warnsdorff, Des Rbsselsprunges einfachste und allgemeinste Losung, 

 Schmalkaldcn, 1823 : Bee also Jaenisch, vol. n, pp. 66 — 61, 273—289. 



