Another Book Wrecker

Considering digit-by-digit as follows:

1. Considering only the unit digit , every 10 pages there is a 7. Therefore 1 sheet is torn every 10 pages or 10 sheet torn from page 1 to 100.
2. But when the tenth digit is 7, then additional 5 sheets 69/70, 71/72, 73/74, 75/76. (77/78 has already torn) and 79/80 are torn. Therefore for every 100 pages, $10+5=15$ sheets are torn.
3. But when the hundred digit is 7, sheets 699/700 to 799/800 or 51 sheets less the 15 sheets already torn or (51-15=36) sheets are torn in addition. Therefore from page 1 to page 1000, $10\times 15+36= 186$ sheets are torn.
4. Then from page 1 to page 2000, $186 \times 2 = 372$ sheets are torn. From page 2001 to 2018, sheets 2007/8 and 2017/18, two sheets are torn. Therefore, from page 1 to page 2018, $372+2=\boxed{374}$ sheets are torn.

Just a different approach.

From 2001 - 2018 there are 2 sheets torn out (with page numbers 2007/2008 and 2017/2018). Putting those aside for now, we consider the first 2000 pages.

We start with the 200 pages that have a 7 in the hundreds position, i.e. 700 - 799 and 1700 - 1799. In the first set, most pages are two per sheet, the exceptions being page 700 (which is on sheet 699/700) and page 799 (which is on sheet 799/800), so rather than 50 sheets we have to tear out 51 sheets. The same would be true for the second set, so we're tearing out a total of 102 sheets.

Now we have 2000 - 200 = 1800 pages left. Ten out of every hundred (for a total of 180 pages) have a 7 in the tens position, i.e 70 - 79, 170 - 179, ... , 1970 - 1979. Like with the hundreds, most pages are two per sheet, except the first and last in every set of ten; so for every set of ten we have to tear out 6 pages (rather than 5). Since there are 18 sets of ten pages, we tear out 18 x 6 = 108 sheets.

We still have 1800 - 180 = 1620 pages left. One in every ten (for a total of 162 pages) have a 7 in the ones position, i.e. 7, 17, 27, ... , 1997. Unlike the previous two batches, these are all one per sheet, so we have to tear out 162 sheets.

Adding these up, and not forgetting the two sheets we removed at the very beginning, our total is 102 + 108 + 162 + 2 = $\boxed{374}$ sheets.

We consider the numbers $1-100$ first.

Among them, the numbers $7,17,27,37,47,57,67,70,71,72,73,74,75,76,77,78,79,87,97$ are the numbers containing the digit $7$ .

However, do take note that the pairs $\underbrace{69,70},\underbrace{71,72},\underbrace{73,74},\underbrace{75,76},\underbrace{77,78},\underbrace{79,80}$ are on the same sheets of paper.

So here, $\cfrac{79-69}{2}+1=6$ sheets of paper have been torn off.

The remaining are $7,17,27,37,47,57,67,87,97$ , and there are $9$ sheets of paper torn off here.

Summing up, we have $15$ sheets of paper torn in the first $100$ pages.

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Now, we move on to consider the numbers $1-1000$ .

Notice that the intervals $1-100,101-200,201-300,301-400,401-500,501-600,601-698,801-900,901-1000$ all have the same number of torn off sheets. This is because the first digit is not $7$ .

So here, we have $15\cdot 9=135$ torn sheets of paper.

As for $\underbrace{699,700},\underbrace{701,702},...,\underbrace{797,798},\underbrace{799,800}$ , there are $\cfrac{799-699}{2}+1=51$ torn sheets of paper here.

Summing up, we have $186$ sheets of paper torn in the first $1000$ pages.

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We move on to consider the numbers $1-2000$ .

Notice that the intervals $1-1000,1001-2000$ all have the same number of torn off sheets. Again, this is because the first digit is not $7$ .

Summing up, we have $186\cdot 2=372$ sheets of paper torn in the first $2000$ pages.

After $2000$ ,

$2007, 2017$ are the numbers which have the digit $7$ in them.

So, in total, there are $372+2=374$ sheets of paper torn off from the book which has $2018$ non-blank pages.