12 ARITHMETICAL RECREATIONS [CH. I 



found by the following rule, (i) Take one of the numbers, say, a, 

 and multiply it by 2. (ii) Add 3 to the product, (iii) Multiply 

 this by 5, and add 7 to the product, (iv) To this sum add the 

 second number, b. (v) Multiply the result by 2. (vi) Add 3 

 to the product, (vii) Multiply by 5, and, to the product, add 

 the third number, c. The result is 100a+10& + c + 235. Hence, 

 if the final result is known, it is sufficient to subtract 235 from 

 it, and the remainder will be a number of three digits. These 

 digits are the numbers chosen originally. 



Third Example*. The following rule for finding the age 

 of a man born in the 19th century is of the same kind. Take 

 the tens digit of the year of birth ; (i) multiply it by 5 ; (ii) to 

 the product add 2 ; (iii) multiply the result by 2 ; (iv) to this 

 product add the units digit of the birth-year ; (v) subtract the 

 sum from 120. The result is the man's age in 1916. 



The algebraic proof of the rule is obvious. Let a and b be 

 the tens and units digits of the birth-year. The successive opera- 

 tions give (i) 5a; (ii) 5a + 2; (iii) 10a + 4; (iv) 10a + 4 + b; 

 (v) 120 - (10a + 6), which is his age in 1916. The rule can be 

 easily adapted to give the age in any specified year. 



Fourth Example^. Another such problem but of more 

 difficulty is the determination of all numbers which are in- 

 tegral multiples of their reversals. For instance, among 

 numbers of four digits, 8712 = 4 x 2l78,.and 9801 = 9 x 1089 

 possess this property. 



Other Problems with numbers in the denary scale. 

 I may mention here two or three other problems which seem 

 to be unknown to most compilers of books of puzzles. 



First Problem. The first of them is as follows. Take any 

 number of three digits, the first and third digits being different : 

 reverse the order of the digits : subtract the number so formed 

 from the original number : then, if the last digit of the difference 

 is mentioned, all the digits in the difference are known. 



* A similar question was given by Laisant and Perrin in their Algebre, Paris, 

 1892 ; and in L' Illustration for July 13, 1895. 



t L'Intermidiaire des MatMmaticiem, Paris, vol. xv, 1908, pp. 228, 278* 

 vol. xvi, 1909, p. 34; vol. xrx, 1912, p. 128. 



