288 CALCULATING PRODIGIES [CH. XIII 



consists of a number of two digits. Knowing the fifth powers 

 to 10, 20,... 90 we have, in order to know the first digit of the 

 answer, only to see between which of these powers the number 

 proposed lies, and the last digit being obvious we can give the 

 answer instantly. If the number is higher, but less than 10 15 , 

 the answer is a number of three digits, of which the middle 

 digit can be found almost as rapidly as the others. This is 

 rather a trick than a matter of mental calculation. 



In his later exhibitions, Bidder was sometimes asked to 

 extract roots, correct to the nearest integer, the exact root 

 involving a fraction. If he suspected this he tested it by 

 " casting out the nines," and if satisfied that the answer was 

 not an integer proceeded tentatively as best he could. Such 

 a question, if the answer is a number of three or more digits, 

 is a severe tax on the powers of a mental calculator, and is 

 usually disallowed in public exhibitions. 



Colburn's remarkable feats in factorizing numbers led to 

 similar questions being put to Bidder, and gradually he evolved 

 some rules, but in this branch of mental arithmetic I do not 

 think he ever became proficient. Of course a factor which is a 

 power of 2 or of 5 can be obtained at once, and powers of 3 can 

 be obtained almost as rapidly. For factors near the square root 

 of a number he always tried the usual method of expressing the 

 number in the form a" — 2> 2 , in which case the factors are obvious. 

 For other factors he tried the digital method already described. 



Bidder was successful in giving almost instantaneously the 

 answers to questions about compound interest and annuities: 

 this was peculiar to him, but his method is quite simple, and 

 may be illustrated by his determination of the compound 

 interest on £100 at 5 per cent, for 14 years. He argued that 

 the simple interest amounted to £(14) (5), i.e. to £70. At the 

 end of the first year the capital was increased by £5, the annual 

 interest on this was 5s. or one crown, and this ran for 13 years, 

 at the end of the second year another £5 was due, and the 5s. 

 interest on this ran for 12 years. Continuing this argument he 

 had to add to the £70 a sum of (13 + 12 + ... + 1) crowns, i.e. 

 (13/2) (14) (5) shillings, i.e. £22. 15s. 04, which, added to the 



