134 CHESS-BOARD RECREATIONS [OH. VI 



If we start from any of the cells mentioned above, the rook 

 takes sixteen moves. If we start from any cell in the middle 

 of one of these moves, it will take seventeen moves to cover 

 this route, but I believe that in most cases wherever the initial 

 cell be chosen sixteen moves will suffice, though in general the 

 route will not be symmetrical. On a board of n" cells it is 

 possible to find a route by which a rook can move successively 

 from its initial cell to every other cell once and only once. 

 Moreover* starting on any cell its path can be made to termi- 

 nate, if ji be even, on any other cell of a different colour, and, 

 if w be odd, on any other cell of the same colour. 



Bishop's Re-Entrant Path. As yet another instance, a bishop 

 can traverse all the cells of one colour on the board in seven- 

 teen- moves if the initial cell is properly chosenf; for instance, 

 starting from the cell 11, it may move successively to the cells 

 55, 82, 71, 17, 28, 46, 13, 31, 86, 68, 57, 48, 15, 51, 84, 66, 88. 

 One more move will bring it back to the initial cell. From 

 the nature of the case, it must traverse some cells more than 

 once. 



Miscellaneous Problems. We may construct numerous such 

 problems concerning the determination of routes which cover 

 the whole or part of the board subject to certain conditions. 

 I append a few others which may tax the ingenuity of those 

 not accustomed to such problems. 



Routes on a Chess-Board. One of the simplest is the 

 determination of the path taken by a rook, placed in the cell 

 11, which moves, one cell at a time, to the cell 88, so that in 

 the course of its path it enters every cell once and only once. 

 This can be done, though I have seen good mathematicians 

 puzzled to effect it. A hasty reader is apt to misunderstand 

 the conditions of the problem. 



Another simple problem of this kind is to move a queen from 

 the cell 33 to the cell 66 in fifteen moves entering every cell once 



* L'Intermediaire des Mathematiciens, Paris, 1901, vol. vm, pp. 163 — 154. 

 t H. E. Dudeney, The Tribune, Deo. 3, 1906. 



