CH. Vl] CHESS-BOARD RECREATIONS 129 



The above methods have been applied to boards of various 

 shapes, especially to boards in the form of rectangles, crosses, 

 and circles*. 



All the more recent investigations impose additional restric- 

 tions : such as to require that the route shall be re-entrant, or 

 more generally that it shall begin and terminate on given cells. 



The simplest solution with which I am acquainted is due to 

 De Lavernede, but is more generally associated with the name 

 of Roget whose paper in 1840 attracted general notice to it f. 

 It divides the whole route into four circuits, which can be 

 combined so as to enable us to begin on any cell and termi- 

 nate on any other cell of a different colour. Hence, if we like 

 to select this last cell at a knight's move from the initial cell, 

 we obtain a re-entrant route. On the other hand, the rule 

 is applicable only to square boards containing (4w) 2 cells: for 

 example, it could not be used on the board of the French jeu 

 des dames, which contains 100 cells. 



Roget began by dividing the board of 64 cells into four 

 quarters. Each quarter contains 16 cells, and these 16 cells 

 can be arranged in 4 groups, each group consisting of 4 cells 

 which form a closed knight's path. All the cells in each such 

 path are denoted by the same letter I, e, a, or p, as the case 

 may be. The path of 4 cells indicated by the consonants I and 

 the path indicated by the consonants p are diamond-shaped: 

 the paths indicated respectively by the vowels e and a are 

 square-shaped, as may be seen by looking at one of the four 

 quarters in figure i below. 



Now all the 16 cells on a complete chess-board which are 

 marked with the same letter can be combined into one circuit, 

 and wherever the circuit begins we can make it end on any 

 other cell in the circuit, provided it is of a different colour 

 to the initial cell. If it is indifferent on what cell the 

 circuit terminates we may make the circuit re-entrant, and 

 * See ex. gr. T. Cicoolini's work Del Gavallo degli Scacchi, Paris, 1836. 

 t J. E. T. de Lavernede, MSmoires de VAcadimie Royale du Gard, Nimes, 

 1839, pp. 151 — 179. P. M. Roget, Philosophical Magazine, April, 1840, series 3, 

 vol. xvi, pp. 305 — 309 ; see also the Quarterly Journal of Mathematics for 1877, 

 vol. xiv, pp. 354—359 ; and the Leisure Hour, Sept. 13, 1873, pp. 587—590, and 

 Dee. 20, 1873, pp. 813—815. 



B. B. 9 



