CH. Vl] 



CHESS-BOARD RECREATIONS 



133 



Analogous Problems. Similar problems can be constructed 

 in which it is required to determine routes by which a piece 

 moving according to certain laws (ex. gr. a chess-piece such as 

 a king, &c.) can travel from a given cell over a board so as to 

 occupy successively all the cells, or certain specified cells, once 

 and only once, and terminate its route in a given cell. 

 Euler's method can be applied to find routes of this kind: 

 for instance, he applied it to find a re-entrant route by which 

 a piece that moved two cells forward like a castle and then one 

 cell like a bishop would occupy in succession all the black cells 

 on the board. 



King's Re-Entrant Path. As one example here is a re- 

 entrant tour of a king which moves successively to every cell 



King's Magic Tour on a Chess-Board. 



of the board. I give it because the numbers indicating the 

 cells successively occupied form a magic square. Of course this 

 also gives a solution of a re-entrant route of a queen covering 

 the board. 



Rook's Re-Entrant Path. There is no difficulty in con- 

 structing re-entrant tours for a rook which moves successively 

 to every cell of the board. For instance, if the rook starts from 

 the cell 11 it can move successively to the cells 18, 88, 81, 71, 

 77, 67, 61, 51, 57, 47, 41, 31, 37, 27, 21,- and so back to 11: 

 this is a symmetrical route. Of course this also gives a solution 

 of a re-entrant route for a king or a queen covering the board. 



