You have 2 boxes and an even number ($2n$) of balls in the first box. Your goal is to distribute the balls equally into the two boxes, so that each box contains $n$ balls. You must obey the following protocol:
Step 1, move exactly 1 ball from one box to the other box.
Step 2, move exactly 2 balls from one box to the other box.
Step 3, move exactly 3 balls from one box to the other box.
Continue until you come to step $k$, when you first find that each box contains less than $k$ balls. If this happens, you go back to step 1 and start from there again. So on and so forth.
Example: if $n=5$, a possible plan is (10,0)-(9,1)-(7,3)-(4,6)-(0,10)-(5,5); if $n=2$, you can check that it can't be done.
Question: is there a threshold $N$ for $n$, such that as long as $n\gt N$, you can always find a plan to equally distribute the balls?