Question

The number of digits you have to type to write allthe page numbers of a book starting from1(first page) is 2019. What is the number of pages in that book?​

Answer 1

If we write all the page numbers in a book starting from 1 as a single number like 123456789101112..., the no. of digits of the number thus formed is equal to 2019. Let these page nos. be written in ascending order as followed.

$\underbrace{\sf{123456789101112\dots}}_{\sf{2019\ digits}}$

We have to find no. of pages of the book.

We see there are (9 - 1) + 1 = 9 one - digit numbers, so the no. of digits in our number, formed byone - digit numbers only, is equal to 1 × 9 = 9.

$\sf{\underbrace{\sf{123456789}}_{9\ digits}}$

If we remove these numbers from our number, then our number will have 2019 - 9 = 2010 digits.

$\underbrace{\sf{123456789}}_{\sf{9\ digits}}\underbrace{\sf{101112131415\dots}}_{\sf{2010\ digits}}$

We see there are (99 - 10) + 1 = 90 two - digit numbers, so the no. of digits in our number, formed by two - digit numbers only, is equal to 2 × 90 = 180.

$\underbrace{\sf{10111213\dots979899}}_{\sf{180\ digits}}$

If we remove these numbers from our number too, then our number will have 2010 - 180 = 1830

$\underbrace{\sf{123456789}}_{\sf{9\ digits}}\underbrace{\sf{101112\dots9899}}_{\sf{180\ digits}}\underbrace{\sf{100101102103\dots}}_{\sf{1830\ digits}}$

We see there are (999 - 100) + 1 = 900 three - digit numbers in total, so the no. of digits formed by three - digit numbers only, in general, is equal to 3 × 900 = 2700.

But our number consists of 1830 digits formed by three - digit numbers only. Hence we can say the final page no. is a three digit number. Let it be $\sf{n.}$

We see our number consists of $\sf{(n-100)+1=n-99}$ three digit numbers, so the no. of digits in our number formed by three digit numbers is equal to $\sf{3(n-99),}$ i.e., 1830.

$\underbrace{\sf{100101102103\dots n}}_{\sf{3(n-99)=1830}}$

Hence,

$\longrightarrow\sf{3(n-99)=1830}$

$\longrightarrow\sf{n-99=610}$

$\longrightarrow\sf{\underline{\underline{n=709}}}$

Therefore the book contains $\bf{709}$ pages.