8 ARITHMETICAL RECREATIONS [CH. I 



the sum by, say, c. (iv) Next, tell him to take a/c of the 

 number originally chosen; and (v) to subtract this from the 

 result of the third operation. The result of the first three 

 operations is (na + b)/c, and the result of operation (iv) is na/c : 

 the difference between these is b/c, and therefore is known to 

 you. For example, if a = 6, b = 12, c = 4, then a/c = 1£, and the 

 final result is 3. 



Second Example*. Ask A to take any number of counters 

 that he pleases : suppose that he takes n counters, (i) Ask 

 some one else, say B, to take p times as many, where p is 

 any number you like to choose, (ii) Request A to give q of 

 his counters to B, where q is any number you like to select, 

 (iii) Next, ask B to transfer to A a number of counters equal 

 to p times as many counters as A has in his possession. Then 

 there will remain in B's hands q(p + I) counters: this number 

 is known to you; and the trick can be finished either by 

 mentioning it or in any other way you like. 



The reason is as follows. The result of operation (ii) is 

 that B has pn + q counters, and A has n — q counters. The 

 result of (iii) is that B transfers p(n — q) counters to A : hence 

 he has left in his possession (pn + q) — p(n — q) counters, that 

 is, he has q(p + l). 



For example, if originally A took any number of counters, 

 then (if you chose p equal to 2), first you would ask B to take 

 twice as many counters as A had done ; next (if you chose q 

 equal to 3) you would ask A to give 3 counters to B ; and then 

 you would ask B to give to A a number of counters equal to 

 twice the number then in A's possession ; after this was done 

 you would know that B had 3 (2 + 1), that is, 9 left. 



This trick (as also some of the following problems) may be 

 performed equally well with one person, in which case A may 

 stand for his right hand and B for his left hand. 



Third Example. Ask some one to perform in succession 

 the following operations, (i) Take any number of three digits, 

 in which the difference between the first and last digits exceeds 

 unity, (ii) Form a new number by reversing the order of 



* Bachet, problem xm, p. 123 : Baohet presented the above trick in a form, 

 somewhat more general, but less effective in practice. 



