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I request a hint, clarification (if needed), or your approval (if my doutful solution is correct) for this problem:

Problem 3. Jack tore out several successive pages from a book. The number of the first page he tore out was 183, and it is known that the number of the last page is written with the same digits in some order. How many pages did Jack tear out of the book?

Every hint or clarification you propose is really acknowledged. Here's the link of the book in case you need it.

Here's my doubtful solution:

There are five possibilities of values of the last page: $138$ is forbidden; otherwise, $813$, $831$, $318$, $381$ are valid. Since the verso of the last page must be even, we consider $318$ the number of the last page. Hence Jack tore out 67 pages.

Thank you very much!

P.S. You may note that the book has a section of answers and hints. However, I had the misfortune of carrying the unpleasant feeling of accidentally seeing the solution of an interesting problem. This, of course, ruined the amusement at searching for that solution. This site is not an exception. Moreover, I noticed this same question here; but, the difference between my question and that question posted there is that he gave the answer. Hence I eagerly wish to discuss the problem, and solve it (being given your approval) without seeing the answer. Without seeing the answer and receiveing feedback, I own the possibility of re-thinking my answer, refining it, and improving my mathematical thinking. This is by far the most important exercise for me.

P.S.S. I am not an English-native speaker, so every correction to my writing is really appreciated!

Sophi
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  • "consider the number of the first page $183$ on the obverse or reverse" $;-;$ Odd numbers cannot be on the reverse sides. From wikipedia: "By book publishing convention, the first page of a book, and sometimes of each section and chapter of a book, is a recto page and hence *all recto pages will have odd numbers* and all verso pages will have even numbers". – dxiv Jul 03 '22 at 23:32
  • And anyway, since the first page torn has an odd number and the total number of pages must be even (each sheet torn contains two pages) you must end on an even page. – mau Jul 03 '22 at 23:40
  • Thank you very much for your suggestions! However, @mau, does "ending on an even page" mean that the reverse of that sheet is even? If that is correct, I have corrected my question. – Sophi Jul 04 '22 at 00:03
  • Yes, in a book you never skip page numbers (even if the page is blank), so each sheet has an even and an odd page number. (Then, as @dxiv said, when you open a book you will find the odd page at the right) – mau Jul 04 '22 at 07:20
  • There is an ambiguity in English: when we say a page was torn out of a book, we really mean that a leaf was torn out of the book. Two pages of the book appeared on that leaf. The first leaf Jack tore out had pages $183$ and $184$ printed on it. So when we ask how many pages Jack tore out, do we mean how many leaves it has torn out, or how many pages were printed on those leaves? – David K Oct 01 '22 at 22:52
  • Your answer seems to be counting how many leaves were torn out, but even then it is incorrect: $68$ leaves are torn out. – David K Oct 01 '22 at 22:53
  • The question seems to be missing the assumption that we must tear out two pages at a time. (Which is a pretty weird assumption that the authors/translator should have made clear. Perhaps in the USSR/Russia of the early 1990s, most books were printed in this "format"?) –  Jan 23 '23 at 08:51

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