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CHAPTER XII. 

 miscellaneous problems (continued). 



This book was stereotyped some years ago, and I have ex- 

 plained above that subsequent additions have had to be fitted 

 in where I could best find a place for them. In this chapter 

 I put together a few additional problems for which I have been 

 unable to find room in the chapters in which they would naturally 

 have been included. Preferably the contents of this and the 

 last chapters should have been inserted in the first four chapters, 

 but to avoid such extensive rearrangement of the matter, I print 

 these problems here. 



I begin with Arithmetical Recreations, somewhat different 

 in character to the selected typical examples given in chapter I, 

 taking in succession Digit Questions, Restorations, and Calendar 

 Problems. 



Digit Questions. Certain questions about digits of numbers 

 are, by custom, presented under the title Arithmetical Recrea- 

 tions. Legendre gave a few examples of the kind. 



Here are two time-honoured arid easy instances, interesting 

 because of their ancient lineage. The first of them is as follows : 

 If n digits are required in the pagination of a book, how many 

 pages are contained therein ; for instance, n = 3001 ? 



The analysis is obvious. The first 999 pages require the use 

 of (9 + 180 + 2700) digits. But 112 additional digits are em- 

 ployed, and these suffice to identify 28 more pages. Therefore 

 the total number of pages is 999 + 28, that is, 1027. 



Here is another example on elementary arithmetic. The 

 numbers from 1 upwards are written consecutively. What is 

 the nth digit : for instance, n = 500,000 ? 



The numbers from 1 to 99,999 inclusive require 488,889 digits. 

 Hence we want the 11,111th digit in the series of six-digit 



