10 ARITHMETICAL RECREATIONS [CH. I 



The following analysis explains the rule, and shows that the 

 final result is independent of the sum written down initially. 



£ s. d. 



(i) a b 



(ii) c b a 



(iii) a-c- 1 19 c-a + 12 



(iv) c-a+12 19 a-c- 1 



(v) 11 38 11 



Mr J. H. Schooling has used this result as the foundation 

 of a slight but excellent conjuring trick. The rule can be 

 generalized to cover any system of monetary units. 



Problems involving Two Numbers. I proceed next to 

 give a couple of examples of a class of problems which involve 

 two numbers. 



First Example*. Suppose that there are two numbers, one 

 even and the other odd, and that a person A is asked to select 

 one of them, and that another person B takes the other. It is 

 desired to know whether A selected the even or the odd number. 

 Ask A to multiply his number by 2, or any even number, and B 

 to multiply his by 3, or any odd number. Request them to add 

 the two products together and tell you the sum. If it is even, 

 then originally A selected the odd number, but if it is odd, then 

 originally A selected the even number. The reason is obvious. 



Second Example^. The above rule was extended by Bachet 

 to any two numbers, provided they are prime to one another 

 and one of them is not itself a prime. Let the numbers be 

 m and n, and suppose that n is exactly divisible by p. Ask A 

 to select one of these numbers, and B to take the other. Choose 

 a number prime to p, say q. Ask A to multiply his number by 

 q, and B to multiply his number by p. Request them to add the 

 products together and state the sum. Then A originally selected 

 m or n, according as this result is not or is divisible by p. The 

 numbers, rn = 7, n = 15, p = 3, q = 2, will illustrate the rest. 



Problems depending on the Scale of Notation. Many 

 of the rules for finding two or more numbers depend on the 

 * Bachet, problem ix, p. 107. + Bachet, problem xi, p. 113. 



