x xabababab. 

 b aab abax xb. 

 baabx xaabb. 

 bx xbaaaabb. 

 bbbbaaaax x. 



76 GEOMETRICAL RECREATIONS [CH. IV 



in four moves, each of a pair of two contiguous pieces, without 

 altering the relative position of the pair, to form a continuous 

 line of four halfpence followed by four florins. 



This can be solved as follows. Let a florin be denoted by a 

 and a halfpenny by b, and let x x denote two contiguous blank 

 spaces. Then the successive positions of the pieces may be 

 represented thus: 



Initially .... 



After the first move . 



After the second move 



After the third move . 



After the fourth move 



The operation is conducted according to the following rule. 

 Suppose the pieces to be arranged originally in circular order, 

 with two contiguous blank spaces, then we always move to the 

 blank space for the time being that pair of coins which occupies 

 the places next but one and next but two to the blank space on 

 one assigned side of it. 



A similar problem with In counters — n of them being white 

 and n black — will at once suggest itself, and, if n is greater 

 than 4, it can be solved in n moves. I have however failed to 

 find a simple rule which covers all cases alike, but solutions, due 

 to M. Delannoy, have been given* for the four cases where n is 

 of the form 4m, 4m + 2, 4sm + 1, or 4m + 3 ; in the first two cases 

 the first \n moves are of pairs of dissimilar counters and the 

 last \n moves are of pairs of similar counters ; in the last two 

 cases, the first move is similar to that given above, namely, 

 of the penultimate and antepenultimate counters to the be- 

 ginning of the row, the next \{n — 1) moves are of pairs of 

 dissimilar counters, and the final \ (n — 1) moves are of similar 

 counters. 



The problem is also capable of solution if we substitute the 

 restriction that at each move the pair of counters taken up must 

 be moved to one of the two ends of the row instead of the 

 condition that the final arrangement is to be continuous. 



* La Nature, June, 1887, p. 10. 



