Boring Book

After finishing a very boring book, Fred notices that the page numbers contained exactly 29 zeros and exactly 137 ones. The book started with page 1. How many pages were in the book?
Source: NCTM Mathematics Teacher, December 2005

We arrange the numbers in rows of 10 numbers as follows
Each row starting from row 10 contains 1 zero except row 100 which contains 11 zeros. The number of rows containing 1 zero is 18. The total number of zeros from page 1 to page 199 is
18\times 1+1\times 11=29
Thus, the book has at most 199 pages.
On the other hand, 9 rows have 1 one, 10 rows have 11 ones, while row 110 has 21 ones. The total number of ones from page 1 to page 199 is
9\times 1+10\times 11+1\times 21=140
Since the book has 137 ones, the last page must be 196. The book has 196 pages.

Answer: 196


About mvtrinh

Retired high school math teacher.
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