CH. i] AEITHMET1CAL KECREATIONS 23 



he would take n h k counters. The total number of counters 

 taken is 2,n h k. Divide this by n, then the remainder will be 

 the number on the domino selected by P ; divide the quotient 

 by n, and the remainder will be the number on the domino 

 selected by P x ; divide this quotient by n, and the remainder 

 will be the number on the domino selected by P 2 ; and so on. 

 In other words, if the number of counters taken is expressed in 

 the scale of notation whose radix is n, then the (h + l)th digit 

 from the right will give the number on the domino selected byP A . 



Thus in Bachet's problem with 3 people and 3 dominoes, 

 we should first give one counter to Q, and two counters to R, 

 while P would have no counters; then we should ask the 

 person who had selected the domino marked or a to take 

 as many counters as he had already, whoever had selected the 

 domino marked 1 or e to take three times as many counters as 

 he had already, and whoever had selected the domino marked 2 

 or i to take nine times as many counters as he had already. 

 By noticing the original number of counters, and observing 

 that 3 of these had been given to Q and R, we should know 

 the total number taken by P, Q, and R. If this number were 

 divided by 3, the remainder would be the number of the 

 domino chosen by P ; if the quotient were divided by 3 the re- 

 mainder would be the number of the domino chosen by Q; and the 

 final quotient would be the number of the domino chosen by R. 



Exploration Problems. Another common question is con- 

 cerned with the maximum distance into a desert which could 

 be reached from a frontier settlement by the aid of a party of 

 n explorers, each capable of carrying provisions that would last 

 one man for a days. The answer is that the man who reaches 

 the greatest distance will occupy na/(n + 1) days before he 

 returns to his starting point. If in the course of their journey 

 they may make dep6ts, the longest possible journey will occupy 

 \a (1 + \ + § + ... + 1/n) days. 



The Josephus Problem. Another of these antique problems 

 consists in placing men round a circle so that if every mth man 

 is killed, the remainder shall be certain specified individuals. 

 Such problems can be easily solved empirically. 



