The Book Job!

Sum of (Number of digits × Number of numbers with those many digits)

$=( 1×9) + (2×90) + 3(900) = 2889$

2929-2889= 40. Thus there were 40 more digits printed, and all of them made 4-digit numbers, hence an extra $\frac {40}{4}=10$ numbers were printed.

Hence the total number of numbers/pages = $9+90+900+10=1009$

Let $d(n)$ be the number of digits printed on the first $n$ pages of the book. Then

• From page 1 to page 9, each page is printed with 1 digit therefore $d(9) = 1(10-1) = 9$ .
• From page 10 to page 99, each page is printed with 2 digits therefore $d(99) - d(9) = 2(10^2-10) = 180$ $\implies d(99) = 9 + 180 = 189$
• From page 100 to page 999, each page is printed with 3 digits therefore $d(999) - d(99) = 3(10^3-10^2) = 2700$ $\implies d(999) = 189 + 2700 = 2889$
• For $d(n) = 2929$ , where we need to find $n$ , we note that $d(n) - d(999) = 2929 - 2889 = 40$ , That is 40 digits more to be printed in the remaining $n-999$ pages. Since each of these pages are printed with 4 digits, we have $4(n-999) = 40$ $\implies n - 999 = 10$ $\implies n = 999+10 = \boxed{1009}$ .

1. Our book needed 2,929 digits to be stamped. Start by getting a feel for how many pages this might be. To do this it’s good to think about the pages in sequence, from the beginning, thinking about how many digits there are in each page number.
2. Pages 1-9 have one digit each, so the total number of digits to be stamped for these pages is 9. This means our book has more than 9 pages.
3. Pages 10-99 all need two digits to be stamped. From pages 10 to 99 inclusive there are 90 pages. So, for these 90 pages 90 x 2 = 180 digits need to be stamped. This means that from page 1 – 99, 189 digits need to be stamped. So our book has more than 99 pages.
4. Pages 100-999 all need three digits to be stamped. There are 900 pages from page 100 to page 999 inclusive and so 900 x 3 = 2700 digits need to be stamped. This means that from page 1 – 999, 2889 digits need to be stamped. So our book has more than 999 pages.
5. Our book has 2,929 digits stamped. How many more than 2,889 is this? This is just 2,929 – 2,889 = 40.
6. Pages 1000, 1001, 1002 now all have 4 digits. This must mean that there are 40/4 = 10 more pages after page 999.
7. This means our book has 1009 pages.