14 ARITHMETICAL RECREATIONS [CH. I 



number of times) is also permitted, we can get to 264: if 

 negative integral indices are also permitted, to 276; and if 

 fractional indices are permitted, to 312. Many similar questions 

 may be proposed, such as using four out of the digits 1, 2, 3, 4, 5. 

 With the five digits 1, 2, 3, 4, 5, each being used once and only 

 once, I have got to 3832 and 4282, according as negative and 

 fractional indices are excluded or allowed. 



Four Fours Problem. Another traditional recreation is, 

 with the ordinary arithmetic and algebraic notation, to express 

 the consecutive numbers from 1 upwards as far as possible in 

 terms of four "4's." Everything turns on what we mean by 

 ordinary notation. If (a) this is taken to admit only the use of 

 the denary scale (ex. gr. numbers like 44), decimals, brackets, 

 and the symbols for addition, subtraction, multiplication and 

 division, we can thus express every number up to 22 inclusive. 

 If (#) also we grant the use of the symbol for square root 

 (repeated if desired a finite number of times) we can get to 

 30 ; but note that though by its use a number like 2 can be 

 expressed by one " 4," we cannot for that reason say that "2 is 

 so expressible. If (7) further we permit the use of symbols for 

 factorials we can express every number to 112. Finally, if (S) 

 we sanction the employment of integral indices expressible by 

 a " 4 " or " 4's " and allow the symbol for a square root to be 

 used an infinite number of times we can get to 156 ; but if (e) 

 we concede the employment of integral indices and the use of 

 sub-factorials* we can get to 877. These interesting problems 

 are typical of a class of similar questions. Thus, under conditions 

 7 and using no indices, with four " l's " we can get to 34, with 

 four " 2's " to 36, with four " 3's " to 46, with four " 5's " to 36, 

 with four " 6's" to 30, with four " 7's" to 25, with four "8's" to 

 36, and with four " 9's " to 66. 



Problems with a series of things which are numbered. 

 Any collection of things numbered consecutively lend themselves 

 to easy illustrations of questions depending on elementary pro- 

 perties of numbers. As examples I proceed to enumerate a 

 few familiar tricks. The first two of these are commonly shown 



* Sub-factorial n is equal to n\ (l-l/l! + l/2!-l/3! + ...±l/n!). On the 

 use of this for the four "4's" problem, see the Mathematical Gazette, May 1912, 



