22 AEITHMETICAL RECREATIONS [CH. I 



more elaborate. Suppose that three people, P, Q, R, select three 

 things, which we may denote by a, e, i respectively, and that it 

 is desired to find by whom each object was selected*. 



Place 24 counters on a table. Ask P to take one counter, 

 Q to take two counters, and R to take three counters. Next, 

 ask the person who selected a to take as many counters as he 

 has already, whoever selected e to take twice as many counters 

 as he has already, and whoever selected i to take four times as 

 many counters as he has already. Note how many counters 

 remain on the table. There are only six ways of distributing 

 the three things among P, Q, and R ; and the number of counters 

 remaining on the table is different for each way. The remainders 

 may be 1, 2, 3, 5, 6, or 7. Bachet summed up the results in 

 the mnemonic line Par fer (1) Cesar (2) jadis (3) devint (5) si 

 grand (6) prince (7). Corresponding to any remainder is a 

 word or words containing two syllables: for instance, to the 

 remainder 5 corresponds the word devint. The vowel in the 

 first syllable indicates the thing selected by P, the vowel in 

 the second syllable indicates the thing selected by Q, and of 

 course R selected the remaining thing. 



Extension. M. Bourlet, in the course of a very kindly 

 notice f of the second edition of this work, gave a much neater 

 solution of the above question, and has extended the problem 

 to the case of n people, P„, P u P 2 , ..., P n - U each of whom 

 selects one object, out of a collection of n objects, such as 

 dominoes or cards. It is required to know which domino or 

 card was selected by each person. 



Let us suppose the dominoes to be denoted or marked by 

 the numbers 0, 1, ..., n — 1, instead of by vowels. Give one 

 counter to P u two counters to P 2 , and generally k counters to 

 Pj. Note the number of counters left on the table. Next 

 ask the person who had chosen the domino to take as many 

 counters as he had already, and generally whoever had chosen 

 the domino h to take n h times as many dominoes as he had 

 already: thus if P^ had chosen the domino numbered h, 



* Bachet, problem xxv, p. 187. 



+ Bulletin des Sciences Mathematiques, Paris, 1893, vol. xvn, pp. 105 — 107. 



