Dr. Roget on the Problem of the Knighfs Move at Chess. 307 



as shown in the central cUagrani in the group marked LEAP, 

 Plate I. and let the squares in each quarter be considered as 

 grouped together into four sets, designated severally by the 

 letters L, E, A, P ; thus, 



L 



E 



A 



P 



A 



P 



L 

 P 

 E 



E 

 A 

 L 



E 

 P 



L 

 A 



It is here to be observed, that each of these sets of squares, 

 marked with the same letter, constitutes a recurrent circuit 

 of four moves of the knight. 



The squares in the other three quarters being similarly 

 designated, as shown in the central diagram already referred 

 to, it will be found that the several sets in each admit of be- 

 ing connected by knight's moves with the corresponding sets, 

 similarly designated in the adjacent quarters. This is shown 

 in the corner diagrams, L, E, A and P, where the con- 

 nexions among the squares of each set are marked by oblique 

 lines joining their centres*. The sets, thus connected, con- 

 stitute four separate systems, of 16 squares each; and it will 

 also be found that these 16 squares are so disposed that the 

 knight may, in each system, perform the circuit of all its 

 squares, beginning from any one given square, and ending 

 at any other of a different colour. A few trials will soon sa- 

 tisfy the learner that, in every case, this may very easily be 

 accomplished, and generally in a great variety of ways. 



It will next be perceived that the knight can always pass 

 from any of the squares^ (excepting those situated at the 

 corners of the board) of one system denoted by a consonant, 

 to those of a system denoted by a vowel, and contrariwise ; 

 (as is shown by the diagonal lines in the four diagrams in- 

 termediate to the former); but not from vowel to vowel, or 

 from consonant to consonant. From the corner squares, the 

 move can only be made to squares belonging to the same 

 system. 



The solution of the proposed problem includes three cases : 



1. If the given initial and terminal squares belong, the one 

 to a system denoted by a consonant, and the other to a sy- 

 stem denoted by a vowel, then, following the order of the 

 letters when arranged in a circle, thus : 



The white squat ca have a circle, and the black a dot in their centres, 



X 2 



