CH. IV] GEOMETRICAL RECREATIONS 75 



putting 7 on 10, then 5 on 2, th?n 3 on 8, then 1 on 6, and 

 lastly 9 on 4 ; or by putting 4 on 1, then 6 on 9, then 8 on 3, 

 then 10 on 5, and- lastly 2 on 7*. 



There is a somewhat similar game played with eight counters, 

 but in this case the four couples finally formed are not equi- 

 distant. Here the transformation will be effected if we move 

 5 on 2, then 3 on 7, then 4 on 1, and lastly 6 on 8. This form 

 of the game is applicable equally to (8 + 2ra) counters, for if we 

 move 4 on 1 we have left on one side of this couple a row of 

 (8 + 2n — 2) counters. This again can be reduced to one of 

 (8 + 2n — 4) counters, and in this way finally we have left eight 

 counters which can be moved in the way explained above. 



A more complete generalization would be tbe case of n 

 counters, where each counter might be moved over the m 

 counters adjacent to it on to the one beyond them. For instance 

 we may place twelve counters in a row and allow the moving 

 a counter over three adjacent counters. By such movements we 

 can obtain four piles, each pile containing three counters. Thus, 

 if the counters be numbered consecutively, one solution can be 

 obtained by moving 7 on 3, then 5 on 10, then 9 on 7, then 12 

 on 8, then 4 on 5, then 11 on 12, then 2 on 6, and then 1 on 2. 

 Or again we may place sixteen counters in a row and allow 

 the moving a counter over four adjacent counters on to the 

 next counter available. By such movements we can get four 

 piles, each pile containing four counters. Thus, if the counters 

 be numbered consecutively, one solution can be obtained by 

 moving 8 on 3, then 9 on 14, then 1 on 5, then 16 on 12, then 

 7 on 8, then 10 on 7, then 6 on 9, then 15 on 16, then 13 on 1, 

 then 4 on 15, then 2 on 13, and then 11 on 6. 



Second Problem with Counters. Another problem f, of a 

 somewhat similar kind, is of Japanese origin. Place four florins 

 (or white counters) and four halfpence (or black counters) 

 alternately in a line in contact with one another. It is required 

 * Note by J. Fitzpatrick to a French translation of the third edition of this 

 work, Paris, 1898. 



+ Bibliotheca Mathematica, 1896, aeries 3, vol. vi, p. 323 ; P. G. Tait, Phila. 

 tophical Magazine, London, January, 1884, series 5, vol. xvu, p. 39; or Collected 

 Scientific Papers, Cambridge, vol. u, 1890, p. 93. 



