CH. I] ARITHMETICAL RECREATIONS 5 



then the remainder is divided by 100, the quotient will be the 

 number thought of originally. 



For let n be the number selected. Then the successive 

 operations applied to it give (i) 5ra; (ii) 5n + 6; (iii) 20n + 24; 

 (iv) 20w + 33 ; (v) lOOw + 165. Hence the rule. 



Third Method*. Request the person who has thought of 

 the number to perform the following operations, (i) To 

 multiply it by any number you like, say, a. (ii) To divide the 

 product by any number, say, b. (iii) To multiply the quotient 

 by c. (iv) To divide this result by d. (v) To divide the final 

 result by the number selected originally, (vi) To add to the 

 result of operation (v) the number thought of at first. Ask for 

 the sum so found : then, if ac/bd is subtracted from this sum, 

 the remainder will be the number chosen originally. 



For, if n was the number selected, the result of the first four 

 operations is to form nacjbd; operation (v) gives ac/bd; and 

 (vi) gives n + (ac/bd), which number is mentioned. But ac/bd 

 is known; hence, subtracting it from the number mentioned, 

 n is found. Of course a, b, c, d may have any numerical values 

 it is liked to assign to them. For example, if a = 12, 6 = 4, 

 c = 7, d = 3 it is sufficient to subtract 7 from the final result in 

 order to obtain the number originally selected. 



Fourth Method^. Ask some one to select a number less 

 than 90. Request him to perform the following operations, 

 (i) To multiply it by 10, and to add any number he pleases, 

 a, which is less than 10. (ii) To divide the result of step (i) 

 by 3, and to mention the remainder, say, b. (iii) To multiply 

 the quotient obtained in step (ii) by 10, and to add any number 

 he pleases, c, which is less than 10. (iv) To divide the result 

 of step (iii) by 3, and to mention the remainder, say d, and 

 the third digit (from the right) of the quotient; suppose 

 this digit is e. Then, if the numbers a, b, c, d, e are known, 

 the original number can be at once determined. In fact, if 

 the number is 9% + y, where x %■ 9 and y rf- 8, and if r is the 



• Bachet, problem v, p. 80. 



t Educational Times, London, May 1, 1895, vol. XLvin, p. 234. This example 

 is said to have been made up by J. Clerk Maxwell in his boyhood: it is in- 

 teresting to note how widely it differs from the simple Bachet problems pre- 

 viously mentioned. 



