Question 149
Let $n$ and $T$ be positive integers. James has $4 n$ marbles with weights $1,2, \ldots, 4 n$. He places them on a balance scale, so that both sides have equal weight. Andrew may move a marble from one side of the scale to the other, so that the absolute difference in weights of the two sides remains at most $T$.
Find, in terms of $n$, the minimum positive integer $T$ such that Andrew may make a sequence of moves such that each marble ends up on the opposite side of the scale, regardless of how James initially placed the marbles.
Find, in terms of $n$, the minimum positive integer $T$ such that Andrew may make a sequence of moves such that each marble ends up on the opposite side of the scale, regardless of how James initially placed the marbles.