# Help me solve my father’s riddle and get my book back

My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty.
So today before leaving home he sneaked into my room and took the book I am currently reading!

The book is quite old and damaged with one or two pages torn out and as I checked my phone in the morning I find a message along the lines of this:

[A picture of him proudly grinning and holding a torn out page in his hand]

Dear Levix, if you want to know where your book lies then tell me: What page am I holding when the sum of all remaining page numbers (without those 2 he is holding) is equal to $81707$? 🙂

Can anybody provide any advice? (it would be awesome if we could find a general solution to stick it to the man for good. 😉 )

Update: First, I want to thank you all for your kind effort and for helping me out so rapidly! I enjoyed your intelligible answers so much that I couldn’t resist to use this knowledge against him 🙂 The final response I gave was that If the sum of all remaining page numbers had been my birthday than the last 2 digits + 10 (32 41, 32 42) would have added up to the
page numbers of the turn out page he was holding. I not only got my book back – I also received a great big hug. So thank you!

(Pluspoints if you can calculate my birthday)

The book contains $p$ sheets (leafs) and has therefore pagenumbers from $1$ to $2p$.
The sum of all the pagenumbers is then given by

The father holds the page with page number $n$ in his hand, so we need to solve

As $81,707 \le p \Big( 2p + 1 \Big)$, we obtain

but as $n \le 2p$, we obtain

whence

so the book contains $202$ pages, whence the page number is given by

The question is: if the father is holding a page $x$ does that mean to exclude the pagenumbers on both sides of the page?

Then the page that you father is holding is $51/52$.

Hope you get your book back!