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

SOLUTION
We arrange the numbers in rows of 10 numbers as follows
1,2,3,\cdots,9
10,11,12,13,\cdots,19
20,21,22,23,\cdots,29
30,31,32,33,\cdots,39
\cdots
90,91,92,93,\cdots,99
100,101,102,103,\cdots,109
110,111,112,113,\cdots,119
\cdots
180,181,182,183,\cdots,189
190,191,192,193,\cdots,199
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

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s