34 ARITHMETICAL RECREATIONS [CH. II 



The Menage Problem*. Another difficult permutation 

 problem is concerned with the number x of possible arrange- 

 ments of n married couples, seated alternately man and woman, 

 round a table, the n wives being in assigned positions, and the 

 n husbands so placed that a man does not sit next to his wife. 



The solution involves the theory of discordant permutations f, 

 and is far from easy. I content myself with noting the results 

 when n does not exceed 10. When n = 3, x = 1 ; when n = 4, 

 a? = 2 ; when n = 5, x = 13 ; when n = 6, x = 80 ; when n = 7, 

 # = 579; when n = 8, # = 4738; when ra=9, # = 43387; and 

 when w = 10, #=439792. 



Bachet's Weights Problem J. It will be noticed that a 

 considerable number of the easier problems given in the last 

 chapter either are due to Bachet or were collected by him in 

 his classical Problemes. Among the more difficult problems 

 proposed by him was the determination of the least number of 

 weights which would serve to weigh any integral number of 

 pounds from 1 lb. to 40 lbs. inclusive. Bachet gave two 

 solutions : namely, (i) the series of weights of 1, 2, 4, 8, 16, 

 and 32 lbs. ; (ii) the series of weights of 1, 3, 9, and 27 lbs. 



If the weights may be placed in only one of the scale-pans, 

 the first series gives a solution, as had been pointed out in 

 1556 by Tartagliag. 



Bachet, however, assumed that any weight might be placed 

 in either of the scale-pans. In this case the second series gives 

 the least possible number of weights required. His reasoning 

 is as follows. To weigh 1 lb. we must have a 1 lb. weight. To 

 weigh 2 lbs. we must have in addition either a 2 lb. weight or 

 a 3 lb. weight ; but, whereas with a 2 lb. weight we can weigh 

 1 lb., 2 lbs., and 3 lbs., with a 3 lb. weight we can weigh 

 1 lb., (3 - 1) lbs., 3 lbs., and (3 + 1) lbs. Another weight of 

 9 lbs. will enable us to weigh all weights from 1 lb. to 13 lbs. ; 

 and we get thus a greater range than is obtainable with any 



* TMorie des Nombres, by E. Lucas, Paris, 1891, pp. 215, 491—495. 

 + See P. A. MaoMahon, Combinatory Analysis, vol. i, Cambridge, 1915, 

 pp. 253—256. 



} Bachet, Appendix, problem v, p. 215. 



§ Trattato de' numeri e misure, Venice, 1556, vol. n, bk. i, chap, xvi, art. 32. 



