CH. Vl] CHESS-BOARD RECREATIONS 135 



and only once, and never crossing its own track or entering a 

 cell more than once*. 



A somewhat similar, but more difficult, question is the 

 determination of the greatest distance which can be travelled 

 by a queen starting from its own square in five consecutive 

 moves, subject to the condition that it never crosses its own 

 track or enters a cell more than oncef. In calculating the 

 distance it may be assumed that the paths go through the 

 centres of the cells. If the length of the side of a cell is one 

 inch, the distance exceeds 3397 inches. 



Another familiar problem can be enunciated as follows. 

 Construct a rectangular board of mn cells by ruling m + 1 

 vertical lines and n + 1 horizontal lines. It is required to 

 know how many routes can be taken from the top left-hand 

 corner to the bottom right-hand corner, the motion being along 

 the ruled lines and its direction being always either vertically 

 downwards or horizontally from left to right. The answer is 

 the number of permutations of m + n things, of which m are 

 alike of one kind and n are alike of another kind: this is equal to 

 (m + n)\ jmlnl. Thus on a square board containing 16 cells 

 (i.e. one-quarter of a chess-board), where m = n = 4<, there are 70 

 such routes ; while on a common chess-board, where m = n = 8, 

 there are no less than 12870 such routes. A rook, moving ac- 

 cording to the same law, can travel from the top left-hand cell 

 to the bottom right-hand cell in (m + n — 2)!/(m— 1)! (n — 1)! 

 ways. Similar theorems can be enunciated for a parallelo- 

 piped. 



Another question of this kind is the determination of the 

 number of closed routes through mn points arranged in m rows 

 and n columns, following the lines of the quadrilateral net- work, 

 and passing once and only once through each point J. 



Ouarini's Problem. One of the oldest European problems 

 connected with the chess-board is the following which was 



* H. E. Dudeney, The Tribune, Oct. 3, 1906. 

 + Ibid., Oct. 2, 1906. 



% See C. F. Sainte-Marie in L' InterrnSdiaire des MaihSmaticiem, Paris, 

 vol. xi, March, 1904, pp. 86—88. 



