CH. IV] 



GEOMETRICAL RECREATIONS 



79 



is the case the pieces in the row containing the vacant cell 

 can be interchanged. To interchange the pieces in each of 

 the seven rows will require 15 moves. Hence to interchange all 

 the pieces will require 15 + (7 x 15) moves, that is, 120 moves. 



If we place 2n (n + 1) white pawns and 2w (n + 1) black 

 pawns in a similar way on a square board of (2n+l) 2 cells, 

 we can transpose them in 2n(n + 1) (n + 2) moves : of these 

 4re (w + 1) are simple and 2re a (n + 1) are leaps. 



Third Problem with Pawns. The following analogous 

 problem is somewhat more complicated. On a square board 

 of 25 cells, place eight white pawns or counters on the cells 



denoted by small letters in the annexed diagram, and eight 

 black pawns or counters on the cells denoted by capital letters : 

 the cell marked with an asterisk (*) being left blank. Each 

 pawn can move according to the laws already explained — the 

 white pawns being able to move only horizontally from left 

 to right or vertically downwards, and the black pawns being 

 able to move only horizontally from right to left or vertically 

 upwards. The object is to get all the white pawns in the 

 places initially occupied by the black pawns and vice versa. 

 No moves outside the dark line are permitted. 



Since there is only one cell on the board which is unoccupied, 

 and since no diagonal moves and no backward moves are 

 permitted, it follows that at each move not more than two 

 pieces of either colour are capable of moving. There are how- 

 ever a very large number of empirical solutions. In previous 

 editions I have given a symmetrical solution in 48 moves, but 

 the following, due to Mr H. E. Dudeney, is effected in 46 moves : 

 Hhg*Ffc*CBHh*GDFJehbag*GABHEFfdg'Hhbc*CFf*GHh* 



