CH. XI] 



MISCELLANEOUS PROBLEMS 



225 



cell or small square empty; the puzzle is to move them so 

 that finally they occupy the position shown in the first of the 

 annexed figures. 



D Top of Box ft 



A Bottom of Box 



B 



9 



P sjg] pi W. 



We may represent the various stages in the game by sup- 

 posing that the blank space, occupying the sixteenth cell, is 

 moved over the board, ending finally where it started. 



The route pursued by the blank space may consist partly of 

 tracks followed and again retraced, which have no effect on the 

 arrangement, and partly of closed paths travelled round, which 

 necessarily are cyclical permutations of an odd number of 

 counters. No other motion is possible. 



Now a cyclical permutation of n letters is equivalent to 

 n — 1 simple interchanges ; accordingly an odd cyclical permu- 

 tation is equivalent to an even number of simple interchanges. 

 Hence, if we move the counters so as to bring the blank space 

 back into the sixteenth cell, the new order must differ from 

 the initial order by an even number of simple interchanges. If 

 therefore the order we want to get can be obtained from this 

 initial order only by an odd number of interchanges, the 

 problem is incapable of solution ; if it can be obtained by an 

 even number, the problem is possible. 



Thus the order in the second of the diagrams given 

 above is deducible from that in the first diagram by six 

 interchanges ; namely, by interchanging the counters 1 and 2, 



b. e. 15 



