OH. IIJ 



ARITHMETICAL RECREATIONS 



33 



possible different distributions of hands at bridge or whist with 

 a pack of fifty-two cards. 



Voting Problems. As a simple example on combinations 

 I take the cumulative vote as affecting the representation of 

 a minority. If there are p electors each having r votes of which 

 not more than s may be given to one candidate, and n men 

 are to be elected, then the least number of supporters who can 

 secure the election of a candidate must exceed prj(ns + r). 



The Knights of the Round Table. A far more difficult 

 permutation problem consists in finding as many arrangements 

 as possible of n people in a ring so that no one has the same 

 two neighbours more than once. It is a well-known proposition 

 that n persons can be arranged in a ring in («- l)!/2 different 

 ways. The number of these arrangements in which all the 

 persons have different pairs of neighbours on each occasion 

 cannot exceed (n — l)(ra — 2)/2, since this gives the number of 

 ways in which any assigned person may sit between every 

 possible pair selected from the rest. But in fact it is always 

 possible to determine (ra — l)(w— 2)/2 arrangements in which 

 no one has the same two neighbours on any two occasions. 



Solutions for various values of n have been given. Here 

 for instance (if n = 8) are 21 arrangements* of eight persons. 

 Each arrangement may be placed round a circle, and no one 

 has the same two neighbours on any two occasions. 



The methods of determining these arrangements are lengthy, 



and far from easy. 



• Communicated to me by Mr E. G. B. Bergholt, May, 1906, see The Secretary 

 and The Queen, August, 1906. Mr Dudeney had given the problem for the case 

 when n=6 in 1905, and informs me that the problem has been solved by 

 Mr B. D. Bewley when n is even, and that he has a general method applicable 

 when n is odd. Various memoirs on the subjeot have appeared in the mathe- 

 matical journals. 



B. B. 3 



