CH. VI] CHESS-BOARD RECREATIONS 127 



whose distances are respectively x and y from the opposite 

 sides a complementary cell. Thus the cells (x, y) and (9 — x, 

 9 — y) are complementary, where x and y denote respectively 

 the column and row occupied by the cell. Then in Euler's 

 solution the numbers in complementary cells differ by 32 : for 

 instance, the cell (3, 7) is complementary to the cell (6, 2), the 

 one is occupied by 57, the other by 25. 



Roget's method, which is described later, can be also applied 

 to give half-board solutions. The result is indicated above. 

 The close of Euler's memoir is devoted to showing how the 

 method could be applied to crosses and other rectangular 

 figures. I may note in particular his elegant re-entrant sym- 

 metrical solution for a square of 100 cells. 



The next attempt of any special interest is due to Vander- 

 monde*, who reduced the problem to arithmetic. His idea was 

 to cover the board by two or more independent routes taken 

 at random, and then to connect the routes. He defined the 

 position of a cell by a fraction x/y, whose numerator x is the 

 number of the cell from one side of the board, and whose 

 denominator y is its number from the adjacent side of the 

 board; this is equivalent to saying that x and y are the 

 co-ordinates of a cell. In a series of fractions denoting a 

 knight's path, the differences between the numerators of two 

 consecutive fractions can be only one or two, while the corre- 

 sponding differences between their denominators must be two 

 or one respectively. Also x and y cannot be less than 1 or 

 greater than 8. The notation is convenient, but Vander- 

 monde applied it merely to obtain a particular solution of 

 the problem for a board of 64 cells: the method by which 

 he effected this is analogous to that established by Euler, 

 but it is applicable only to squares of an even order. The 

 route that he arrives at is defined in his notation by the 

 following fractions: 5/5, 4/3, 2/4, 4/5, 5/3, 7/4, 8/2, 6/1, 7/3, 

 8/1, 6/2, 8/3, 7/1, 5/2, 6/4, 8/5, 7/7, 5/8, 6/6, 5/4, 4/6, 2/5, 

 1/7, 3/8, 2/6, 1/8, 3/7, 1/6, 2/8, 4/7, 3/5, 1/4, 2/2, 4/1, 3/3, 

 1/2, 3/1, 2/3, 1/1, 3/2, 1/3, 2/1, 4/2, 3/4, 1/5, 2/7, 4/8, 3/6, 

 * L'Histoiri de VAcademie det Sciences for 1771, Paris, 1774, pp. 566—574 



