CH. I] ARITHMETICAL RECREATIONS 13 



For suppose the number is 100a + 106 + c, then the number 

 obtained by reversing the digits is 100c + 106 + a. The difference 

 of these numbers is equal to (100a + c) — (100c + a), that is, to 

 99 (a — c). But a— cis not greater than 9, and therefore the 

 remainder can only be 99, 198, 297, 396, 495, 594, 693, 792, or 

 891 ; in each case the middle digit being 9 and the digit before 

 it (if any) being equal to the difference between 9 and the last 

 digit. Hence, if the last digit is known, so is the whole of the 

 remainder. 



Second Problem. The second problem is somewhat similar 

 and is as follows, (i) Take any number; (ii) reverse the digits; 

 (iii) find the difference between the number formed in (ii) and 

 the given number ; (iv) multiply this difference by any number 

 you like to name; (v) cross out any digit except a nought; 

 (vi) read the remainder. Then the sum of the digits in the 

 remainder subtracted from the next highest multiple of nine 

 will give the figure struck out. This is clear since the result of 

 operation (iv) is a multiple of nine, and the sum of the digits of 

 every multiple of nine is itself a multiple of nine. This and the 

 previous problem are typical of numerous analogous questions. 



Empirical Problems. There are also numerous empirical 

 problems, such as the following. With the ten digits, 9, 8, 7, 6, 

 5, 4, 3, 2, 1, 0, express numbers whose sum is unity: each digit 

 being used only once, and the use of the usual notations for 

 fractions being allowed. With the same ten digits express 

 numbers whose sum is 100. With the nine digits, 9, 8, 7, 6, 5, 

 4, 3, 2, 1, express numbers whose sum is 100. To the making 

 of such questions there is no limit, but their solution involves 

 little or no mathematical skill. 



Four Digits Problem. I suggest the following problem as 

 being more interesting. With the digits 1, 2, 3, 4, express the 

 consecutive numbers from 1 upwards as far as possible : each of 

 the four digits being used once and only once in the expression 

 of each number. Allowing the notation of the denary scale 

 (including decimals), as also algebraic sums, products, and positive 

 integral powers, we can get to 88. If the use of the symbols 

 for square roots and factorials (repeated if desired a finite 



