Call the marbles that Aybike takes out of boxes "disposable marbles." If Aybike can succeed with $n$ moves, he will have at most $2017n-\frac{n(n-1)}{2}$ disposable marbles, because he will have to leave at least $\frac{n(n-1)}{2}$ total marbles in the boxes he grabbed from so those boxes have differing numbers of marbles. To make the remaining $1000-n$ boxes have differing numbers of marbles, he needs at least $\frac{(1000-n)(999-n)}{2}$ disposable marbles. Thus, $$2017n-\frac{n(n-1)}{2}\ge \frac{(1000-n)(999-n)}{2}$$$$\Longleftrightarrow 4034n-n^2+n\ge n^2-1999n+999000$$$$\Longleftrightarrow 6034n-2n^2\ge 999000$$$$\Longleftrightarrow 3017n-n^2\ge 499500$$$$\Longleftrightarrow n^2-3017n+499500\le 0$$$$\Longleftrightarrow n^2-3017n+\left(\frac{3017}{2}\right)^2\le \left(\frac{3017}{2}\right)^2-499500$$$$\Longleftrightarrow \left(\frac{3017}{2}-n\right)^2\le 1776072.25$$$$\Longleftrightarrow \frac{3017}{2}-n\le \sqrt{1776072.25}$$$$\Longleftrightarrow n\ge \frac{3017}{2}-\sqrt{1776072.25}$$$$n\ge 175.806\dots $$To see that $n=176$ works, take $2018-i$ balls out of the $i$th box for each $1\le i\le 176$, and then put $i$ additional marbles in the $(177+i)$th box for each $0\le i\le 823$. Finally, put $516$ more of these additional balls in box $1000$. Clearly this gives a distinct number of balls in each box, and this is valid because $$(2017+2016+2015+\dots +1842)=(1+2+3+\dots +823)+516$$Thus, the minimum is $\boxed{176}$.

I dont understand, you didnt use that there are 1000 boxes. There must be something wrong

Thanks - I originally misread it as there being $2017$ boxes. It should be fixed now.

I think the min n is 500.Here is the proof If we make n moves there will be at least 1000-n boxes that have never been touched. Also this boxes can take values between 2017 and 2017+n .So n+1>=1000-n. Hence n>=500

The problem definitely wasn't translated great, but I think it means to say that Aybike can distribute the balls he takes out to many different boxes on a single move. Otherwise the problem would be pretty silly and as you said the answer would simply be $500$.