130 



CHESS-BOARD RECREATIONS 



[CH. VI 



in this case we can make the direction of motion round each 

 group (of 4 cells) the same. For example, all the cells 

 marked p can be arranged in the circuit indicated by the 

 successive numbers 1 to 16 in figure ii below. Similarly all 

 the cells marked a can be combined into the circuit indicated 

 by the numbers 17 to 32 ; all the I cells into the circuit 33 to 

 48 ; and all the e cells into the circuit 49 to 64. Each of the 

 circuits indicated above is symmetrical and re-entrant. The 

 consonant and the vowel circuits are said to be of opposite 

 kinds. 



Roget't Solution (i). Roget's Solution (ii). 



The general problem will be solved if we can combine the 

 four circuits into a route which will start from any given cell, 

 and terminate on the 64th move on any other given cell of a 

 different colour. To effect this Roget gave the two following 

 rules. 



First. If the initial cell and the final cell are denoted the 

 one by a consonant and the other by a vowel, take alternately 

 circuits indicated by consonants and vowels, beginning with 

 the circuit of 16 cells indicated by the letter of the initial 

 cell and concluding with the circuit indicated by the letter of 

 the final cell. 



Second. If the initial cell and the final cell are denoted 

 both by consonants or both by vowels, first select a cell, Y, in 

 the same circuit as the final cell, Z, and one move from it, 

 next select a cell, X, belonging to one of the opposite circuits 

 and one move from Y. This is always possible. Then, leaving 



