CH. n] ARITHMETICAL RECREATIONS 85 



weight less than 9 lbs. Similarly weights of 1, 3, 9, and 

 27 lbs. suffice for all weights up to 40 lbs., and weights of 

 1, 3, 3", ..., 3 n_1 lbs. enable us to weigh any integral number 

 of pounds from 1 lb. to (1 + 3 + 3 2 + ... 3"- 1 ) lbs., that is, to 

 £(3»-l)lbs. 



To determine the arrangement of the weights to weigh any 

 given mass we have only to express the number of pounds in 

 it as a number in the ternary scale of notation, except that in 

 finding the successive digits we must make every remainder 

 either 0, 1, or — 1 : to effect this a remainder 2 must be written 

 as 3 — 1, that is, the quotient must be increased by unity, in 

 which case the remainder is — 1. This is explained in most 

 text-books on algebra. 



Bachet's argument does not prove that his result is unique 

 or that it gives the least possible number of weights required. 

 These omissions have been supplied by Major MacMahon, 

 who has discussed the far more difficult problem (of which 

 Bachet's is a particular case) of the determination of all possible 

 sets of weights, not necessarily unequal, which enable us to 

 weigh any integral number of pounds from 1 to n inclusive, 

 (i) when the weights may be placed in only one scale-pan, and 

 (ii) when any weight may be placed in either scale-pan. He 

 has investigated also the modifications of the results which are 

 necessary when we impose either or both of the further condi- 

 tions (a) that no other weighings are to be possible, and (6) that 

 each weighing is to be possible in only one way, that is, is to 

 be unique*. 



The method for case (i) consists in resolving 1 + x + a? + . . . + x n 

 into factors, each factor being of the form 1 + x a + a? a + . . . + x ma ; 

 the number of solutions depends on the composite character of 

 » + 1. The method for case (ii) consists in resolving the expres- 

 sion ar n + x- n+1 + ...+x- 1 + l+x+ ...+x n - 1 + x n into factors, 

 each factor being of the form x'™ + . . . + ar a + 1 + x a + . . . + x ma ; 

 the number of solutions depends on the composite character of 

 2n + l. 



* See his article in the Quarterly Journal of Mathematics, 18S6, vol. xxi, 

 pp. 367 — 373. An account of the method is given in Nature, Dec. 4, 1S90, 

 vol. xlii, pp. 113 — 114. 



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