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)
Best Answer
The book contains $p$ sheets (leafs) and has therefore pagenumbers from $1$ to $2p$. The sum of all the pagenumbers is then given by
$$ \sum_{i=1}^{2p}i =p \Big( 2p + 1 \Big). $$
The father holds the page with page number $n$ in his hand, so we need to solve
$$ 81,707 = p \Big( 2p + 1 \Big) - n. $$
As $81,707 \le p \Big( 2p + 1 \Big)$, we obtain
$$ p \ge 202, $$
but as $n \le 2p$, we obtain
$$ p \Big(2 p + 1 \Big) - 81,707 \le 2 p, $$
whence
$$ p \le 202, $$
so the book contains $202$ pages, whence the page number is given by
$$ 202 \times 405 - 81,707 = 103. $$
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!