OH. IV] GEOMETRICAL RECREATIONS 77 



Tait suggested a variation of the problem by making it 

 a condition that the two coins to be moved shall also be made to 

 interchange places ; in this form it would seem that five moves 

 are required ; or, in the general case, n + 1 moves are required. 



Problems on a Chess-board with Counters or Pawns. The 

 following three problems require the use of a chess-board as well 

 as of counters or pieces of two colours. It is more convenient 

 to move a pawn than a counter, and if therefore I describe them 

 as played with pawns it is only as a matter of convenience and 

 not that they have any connection with chess. The first is 

 characterized by the fact that in every position not more than 

 two moves are possible ; in the second and third problems not 

 more than four moves are possible in any position. With these 

 limitations, analysis is possible. I shall not discuss the similar 

 problems in which more moves are possible. 



First Problem with Pawns*. On a row of seven squares 

 on a chess-board 3 white pawns (or counters), denoted in the 

 diagram by "a"s, are placed on the 3 squares at one end, and 

 3 black pawns (or counters), denoted by " b "s, are placed on the 

 3 squares at the other end — the middle square being left vacant. 

 Each piece can move only in one direction ; the " a " pieces can 

 move from left to right, and the " b " pieces from right to left. 

 If the square next to a piece is unoccupied, it can move on 



a a a bob 



to that ; or if the square next to it is occupied by a piece of the 

 opposite colour and the square .beyond that is unoccupied, then 

 it can, like a queen in draughts, leap over that piece on to the 

 unoccupied square beyond it. The object is to get all the white 

 pawns in the places occupied initially by the black pawns and 

 vice versa. 



The solution requires 15 moves. It may be effected by 

 moving first a white pawn, then successively two black pawns, 

 then three white pawns, then three black pawns, then three 

 white pawns, then two black pawns, and then one white pawn. 

 We can express this solution by saying that if we number the 

 * Lucas, vol. n, part 5, pp. 141 — 143. 



