The page numbers of a book written in a row gives a 216 digit number. How many pages are there in the book?

Let us start from 1-digit numbers of the pages.

Page numbers always start from 1, never 0.

So,

From Page No. 1 to 9, there are 9 digits.

From Page No. 10 to 99, there are (100 – 10) = 90 pages.

We know 2-digit numbers have 2 digits each, so each number has 2 digits.

For instance, 10 has 2-digits.

From Page No. 10 to 11, we have (2 × 2) = 4 digits.

From Page No. 10 to 12, we have (3 × 2) = 6 digits.

…

…

From Page No. 10 to 99, we have (90 × 2) = 180 digits.

So, from Page No. 1 to 99, we have (9 + 180) = 189 digits.

Now, 216 – 189 = 27

We need to find the number of pages, whose digits add up to 27.

We know 3-digit numbers have 3 digits each, so each number has 3 digits.

For instance, 100 has 3-digits.

From Page No. 100 to 101, we have (2 × 3) = 6 digits.

From Page No. 100 to 102, we have (3 × 3) = 9 digits.

…

…

From Page No. 100 to 108. We have (9 × 3) = 27 digits.

So, from Page No. 1 to 108, we have

9 + 180 + 27 = 216

Hence, there are 108 pages in the book.