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Nезнayka took a book from Pilulkin and counted how many digits were needed to number all the pages, starting from the first. He got 100 digits. Could this be possible, or did Nезнayka make a mistake? If it could, how many pages were there?
Notice, the number of the last page is a two-digit number.

## Solution

Under these conditions, the number of the last page is a two-digit number (the total number of digits in all two-digit and one-digit numbers is $9+90 \times 2>100$). However, all one-digit pages will contribute 9 digits, which is an odd number, and adding any number of pages with two-digit numbers will add an even number of digits, thus keeping the total number of digits odd.

## Answer

Nезнayka made a mistake.