CH. Il] ARITHMETICAL RECREATIONS 31 



similarly the probability that all three come down tail is 1/8 ; 

 hence the probability that all the coins come down alike (i.e. 

 either all of them heads or all of them tails) is 1/4. But, of 

 three coins thus thrown up, at least two must come down alike : 

 now the probability that the third coin comes down head is 1/2 

 and the probability that it comes down tail is 1/2, thus the 

 probability that it comes down the same as the other two coins 

 is 1/2 : hence the ' probability that all the coins come down 

 alike is 1/2. I leave to my readers to say whether either of 

 these conflicting conclusions is right, and, if so, which is 

 correct. 



Arithmetical Problems. To the above examples I may add 

 the following standard questions, or recreations. 



The first of these questions is as follows. Two clerks, A and 

 B, are engaged, A at a salary commencing at the rate of (say) 

 £100 a year with a rise of £20 every year, B at a salary 

 commencing at the same rate of £100 a year with a rise of £5 

 every halfryear, in each case payments being made half-yearly ; 

 which has the larger income ? The answer is B ; for in the 

 first year A receives £100, but B receives £50 and £55 as 

 his two half-yearly payments and thus receives in all £105. In 

 the second year A receives £120, but B receives £60 and £65 

 as his two half-yearly payments and thus receives in all £125. 

 In fact B will always receive £5 a year more than A. 



Another simple arithmetical problem is as follows. A hymn- 

 board in a church has four grooved rows on which the numbers 

 of four hymns chosen for the service are placed. The hymn- 

 book in use contains 700 hymns. What is the smallest number 

 of plates, each carrying one digit, which must be kept in stock so 

 that the numbers of any four different hymns selected can be 

 displayed ; and how will the result be affected if an inverted 6 

 can be used for a 9 ? The answers are 86 and 81. What are 

 the answers if a digit is painted on each side of each plate ? 



As another question take the following. A man bets 1/nth 

 of his money on an even chance (say tossing heads or tails 

 with a penny): he repeats this again and again, each time 

 betting 1/nth of all the money then in his possession. If, 

 finally, the number of times he has won is equal to the number 



