CH. I] ARITHMETICAL RECREATIONS 9 



the digits, (iii) Find the difference of these two numbers, 

 (iv) Form another number by reversing the order of the digits 

 in this difference, (v) Add together the results of (iii) and (iv). 

 Then the sum obtained as the result of this last operation will 

 be 1089. 



An illustration and the explanation of the rule are given 

 below. 



(i) 237 100a + 106 + c 



(ii) 732 100c + 10b+a 



(iii) 495 100(a-c-l) + 90 + (10 + c-a) 



(iv) _594 100(10 + c-a)+ 90 + (a-c-l) 



(v) 1089 900 +180 + 9 



The result depends only on the radix of the scale of notation in 

 which the number is expressed. If this radix is r, the result is 

 (r - 1) (r + l) s ; thus if r = 10, the result is 9 x ll 2 , that is, 1089. 

 Similar problems can be made with numbers exceeding 999. 



Fourth Example*. The following trick depends on the 

 same principle. Ask some one to perform in succession the 

 following operations, (i) To write down any sum of money 

 less than £12, in which the difference between tbe number of 

 pounds and the number of pence exceeds unity, (ii) To 

 reverse this sum, that is, to write down a sum of money ob- 

 tained from it by interchanging the numbers of pounds and 

 pence, (iii) To find the difference between the results of 

 (i) and (ii). (iv) To reverse this difference, (v) To add to- 

 gether the results of (iii) and (iv). Then this sum will be 

 £12. 18s. lid 



For instance, take the sum £10. 17s. 5d.; we have 



£ s. d. 



(i) 10 17 5 



(ii) 5 17 10 



(iii) 4 19 7 



(iv) 7 19 4 



(v) 12 18 11 



* Educational Times Reprinte, 1890, vol. lhi, p. 78. 



