CH. Vl] CHESS-BOARD RECREATIONS 131 



out the cells Z and Y, it always will be possible, by the rule 

 already given, to travel from the initial cell to the cell X in 

 62 moves, and thence to move to the final cell on the 64th 

 move. 



In both cases however it must be noticed that the cells in 

 each of the first three circuits will have to be taken in such an 

 order that the circuit does not terminate on a corner, and it 

 may be desirable also that it should not terminate on any of 

 the border cells. This will necessitate some caution. As far 

 as is consistent with these restrictions it is convenient to make 

 these circuits re-entrant, and to take them and every group in 

 them in the same direction of rotation. 



. As an example, suppose that we are to begin on the cell 

 numbered 1 in figure ii above, which is one of those in a 

 p circuit, and to terminate on the cell numbered 64, which is 

 one of those in an e circuit. This falls under the first rule : 

 hence first we take the 16 cells marked p, next the 16 cells 

 marked a, then the 16 cells marked I, and lastly the 16 cells 

 marked e. One way of effecting this is shown in the diagram. 

 Since the cell 64 is a knight's move from the initial cell the 

 route is re-entrant. Also each of the four circuits in the 

 diagram is symmetrical, re-entrant, and taken in the same 

 direction, and the only point where there is any apparent 

 breach in the uniformity of the movement is in the passage 

 from the cell numbered 32 to that numbered 33. 



A rule for re-entrant routes, similar to that of Eoget, has 

 been given by various subsequent writers, especially by De 

 Polignac* and by Laquieref, who have stated it at much 

 greater length. Neither of these authors seems to have been 

 aware of Roget's theorems. De Polignac, like Koget, illustrates 

 the rule by assigning letters to the various squares in the way 

 explained above, and asserts that a similar rule is applicable to 

 all even squares. 



* Comptes Rendus, April, 1861 ; and Bulletin de la Societi MatMmatique de 

 France, 1881, vol. ix, pp. 17—24. 



t Bulletin de la Sociite MatMmatique de France, 1880, vol. vin, pp. 82— 

 102, 132—158. 



9—2 



