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

$$ \sum_{i=1}^{2p}i =p \Big( 2p + 1 \Big). $$

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!

Let $p,p+1$ be the pagenumbers of the pages he is holding. Assume the book has $n$ pages. Then:

$$81707=\sum_{i=1}^ni-(p+p+1)=\frac{n(n+1)}{2}-2p-1.$$

It is clear that $\frac{n(n+1)}{2}$ must be an even number, say $2m.$ So

$$81707=2m-2p-1=2(m-p)-1\implies m-p=40854.$$ That is

$$\frac{n(n+1)}{4}=p+40854,$$ from where

$$n=\frac{-1+\sqrt{1+16(p+40854)}}{2}.$$ Since the number of pages has to be a natural number, $16p+653665$ must be a square. Since $\sqrt{16p+653665}>808$ we have that $\sqrt{16p+653665}=808+k$ for some natural number $k.$ That is,

$$n=\frac{-1+808+k}{2}=\frac{807+k}{2}.$$

For $k=1$ we have $n=404$ and $p=51.$

For $k=2$ (or any even number) $n$ is not an integer.

Since $n(n+1)$ is a multiple of $4$ we have that $n$ is a multiple of four or a multiple of $4$ minus one. So $k$ must be a multiple of $8$ minus/plus one. So, the next case to consider is $k=7.$ In such a case, $n=407$ and $p=660>n=407,$ which is impossible. (The same happens for bigger values of $k,$ which shows that the solution is unique.)

So, the book has pages numbered from $1$ to $404$ and the given pages have numbers $51$ and $52.$

Here's a simple solution to the problem using a bit of programming.

• Let remainingPageSum equal 81707.

• Let a page number counter pageNumber equal 1.

• Let totalPageSum equal pageNumber.

• While totalPageSum is less than remainingPageSum:

  • Increment pageNumber by 1, and

  • Increment totalPageSum by pageNumber.

Now you have totalPageSum equal to 81810. Subtract remainingPageSum from totalPageSum to get the sum of the removed page numbers, 103. The two consecutive page numbers having a sum of 103 will be floor(103 / 2) and ceil(103 / 2), or 51 and 52.

Test it out here.

Let there be $n$ pages, and suppose your father is on page $k$. Then you want $$\sum_{i=k+1}^ni=81707$$ Do you know how to calculate this sum?