4 ARITHMETICAL RECREATIONS [CH. I 



means of disguising the operations indicated, and give merely 

 a bare enumeration of the steps essential to the success of the 

 method used. To the non-mathematician even to-day some of 

 those results seem astonishing, but the secret is at once re- 

 vealed as soon as the question is translated by symbols into 

 mathematical language. 



TO FIND A NUMBER SELECTED BY SOME ONE. There are 



innumerable ways of finding a number chosen by some one, 

 provided the result of certain operations on it is known. I 

 confine myself to methods typical of those commonly used. 

 Any one acquainted with algebra will find no difficulty in 

 framiDg new rules of an analogous nature. 



First Method*, (i) Ask the person who has chosen the 

 number to treble it. (ii) Enquire if the product is even or 

 odd : if it is even, request him to take half of it ; if it is odd, 

 request him to add unity to it and then to take half of it 

 (iii) Tell him to multiply the result of the second step by 3. 

 (iv) Ask how many integral times 9 divides into the latter 

 product: suppose the answer to be n. (v) Then the number 

 thought of was 2n or 2n + 1, according as the result of step (i) 

 was even or odd. 



The demonstration is obvious. Every even number is of 

 the form In, and the successive operations applied to this give 

 (i) 6re, which is even; (ii) £6n = 3n; (iii) 3x3a = 9»; (iv) 

 \%n = n; (v) 2n. Every odd number is of the form 2m + 1, and 

 the successive operations applied to this give (i) 6n+'3, which 

 is odd; (ii) i (6n + 3 + 1) = 3ra + 2 ; (iii) 3(3»+ 2)=9» + 6; 

 (iv) £ (9n + 6) = n + a remainder ; (v) In + 1. These results 

 lead to the rule given above. 



Second Method"]: Ask the person who has chosen the 

 number to perform in succession the following operations, 

 (i) To multiply the number by 5. (ii) To add 6 to the product, 

 (iii) To multiply the sum by 4. (iv) To add 9 to the product 

 (v) To multiply the sum by 5. Ask to be told the result of 

 the last operation : if from this product 165 is subtracted, and 



* Bachet, Problemes, Lyons, 1G24, problem i, p. 53. 

 t A similar rule was given by Bachet, problem iv, p. 74. 



