[ 305 ] 



LI. Description of a Method of moving the Knight over every 

 square of the Chess-board, "joithout going twice over any one ,• 

 commencing at any given square, and ejiding at any other 

 given square of a different colour. By P. M. Roget, M.D., 

 ^SecR.S. 



[Illustrated by Plate I.] 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 

 T^HE problem of carrying the knight, in a course of his 

 "^ own moves, over every square of tlie chess-board, with- 

 out going twice over any square, has engaged the attention 

 and exercised the ingenuity of mathematicians during the 

 last hundred years*. Even the great Euler condescended to 

 put forth a portion of his giant strength in grappHng with 

 the difficulties it presents, and bringing it within the grasp 

 of his powerful analysis f. Vandermonde attempted to con- 

 struct a general algebraic formula for its solution |. Others, 

 confining their efforts to the attainment of mere practical 

 results, have contented themselves with the search of parti- 

 cular methods of resolving the problem in limited cases, and 

 under the simplest conditions only ; such as that of being 

 obliged to commence the journey of the knight from a given 

 square ; one of the corner squares having usually been se- 

 lected for that purpose^. The next step was the contrivance 

 of methods fulfilling a further condition, namely, that the 

 square at which the tour of the knight terminates shall be 

 so situated as to be one move from the square from which it 

 was begun. It is evident that whenever this has been ac- 

 complished, we have obtained a recurring or circular course, 

 which the knight might again traverse by continuous moves : 

 so that such a course gives us the power of commencing with 

 any given square whatsoever, and of traversing through the 

 whole series of 64? squares, until the entire circuit is com- 

 pleted. 



Various circuits of this kind have been devised, and de- 

 scribed in different memoirs, which have, from time to time, 

 been published || ; and the problem, under this form, has been 



*_See Ozanam, Recreations Mathematiques et Physiques, nouvclle edition, 

 Paris.lJoO, torn. i. p. 260, where De Montniort, De Moivre, and De Mairan, 

 are quoted as having treated this subject. 



t Memoires de I' Acadimie de Berlin for 1759, p. 310, 



+ In a paper entitled " Remarques siir les Pi-oblhnes de Situation," in the 

 MtMoires de f Acadhnie Royale des Sciences, 1771, p. 566. 



y £^^ ''" the point at which the problem is left by Ozanam. 



II See Journal of Science and the Arts, iii. 12. March, 1817; and also 

 Edm. Phil. Journal, iv. c>s7, w 236. 



Phil. Mag, S. 3. Vol. 16. ^o, 103. Jpil 18iO. X 



