I am compiling the new BrainBashers world almanac, and it now contains lots more pages.

I know that it takes 333 digits to print the page numbers in sequence.

How many numbered pages does the book have?

How many times does the number 3 appear?

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Hint

Think of the pages from 1-9, and then 10-99.

Answer

There are 147 pages.
The number 3 appears 35 times.

Answer #1

For pages 1-9, there are 9 pages, which is 9 digits.

For pages 10-99, there are 90 pages, which is 180 digits.

This is a total so far of 189, therefore we require another 333-189 = 144 digits, which is another 144 ÷ 3 = 48 pages.

Taking us to 9 + 90 + 48 = 147 pages in total.

Answer #2

From page 1 to 147 we have 15 pages that end in 3:

3, 13, 23, ..., 143

We also have the ten pages that start with 30:

30, 31, 32, ..., 39

Plus the ten pages that start with 130:

130, 131, 132, ..., 139

For a total of 15 + 10 + 10 = 35 number 3's.