Can You Solve This American Mathematics Olympiad Problem?

It’s Actually Easier than You Think

Problem

In a book, 843 digits were used to number all the pages consecutively, starting with 1. How many pages are there in the book?

Pause for a moment and try to solve this problem yourself! 🧠 🧠 ✏️

Solution

Got your answer? Ok! Here’s the solution! 3…2….1..

This is a relatively simple problem, and to find the total number of pages in the book using 843 digits for numbering, we can break it down by the number of digits used for different ranges of page numbers:

The Pages 1 to 9:

Out of all the pages in the book, only the first nine are single-digit numbers.

Therefore, the total digits in these first 9 pages are equal to:

9 x 1 = 9

The Pages 10 to 99:

There are 99—10 + 1= 90 pages, and each uses 2 digits.

Therefore, the total number of digits is 180.

The Pages 100 and Beyond

Each page number from 100 onward up until 1000 has 3 digits.

The total digits from 1 to 99 are 180 + 9 = 189 digits.

The total number of pages are 99.

If we subtract 189 from 843,

843 — 189=654

Since each number from 100 and onwards uses 3 digits, we can divide the difference of 654 by 3 to find the total number of pages from 100 and onwards.

654 / 3 = 218

Thus, the total number of pages in the book is 99 + 218 = 317.

Final Answer

The total number of pages in the book is 317.