78 



GEOMETRICAL RECREATIONS 



[CH. IV 



cells (a term used to describe each of the small squares on a 

 chess-board) consecutively, then initially the vacant space 

 occupies the cell 4 and in the successive moves it will occupy 

 the cells 3, 5, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 3, 5, 4. Of these moves, 

 six are simple and nine are leaps. 



More generally, if we have m white pawns at one end of a 

 row of m + n + 1 cells, and n black pawns at the other end, the 

 arrangement can be reversed in mn + m + n moves, of which 

 m + n are simple and mn are leaps. 



Second Problem with Pawns*. A similar game may be 

 played on a rectangular or square board. The case of a square 

 board containing 49 cells, or small squares, will illustrate this 

 sufficiently : in this case the initial position is shown in the 

 annexed diagram where the " a ''s denote the pawns or pieces 



of one colour, and the "b"s those of the other colour. The 

 " a " pieces can move horizontally from left to right or verti- 

 cally down, and the "b pieces can move horizontally from 

 right to left or vertically up, according to the same rules as 

 before. 



The solution reduces to the preceding case. The pieces 

 in the middle column can be interchanged in 15 moves. In the 

 course of these moves every one of the seven cells in that 

 column is at some time or other vacant, and whenever that 



* Lucas, vol. ii, part 5, p. 144. 



