3. The Galactic Hotel has 100 rooms, which can accommodate $101,102, \cdots, 200$ guests respectively. Currently, there are $n$ guests staying in these rooms. Now a VIP group has arrived, and a whole room needs to be vacated for them. For this purpose, the hotel manager selects one room and arranges all the guests originally staying in it to move to another single room. The question is: For which $n$ can the hotel manager arrange the guests in this way, regardless of their current room assignments?
3. $n \leqslant 8824$.

Let $n=8 \quad 824$.
If the hotel manager cannot arrange the guests as described, pair the rooms: $(101,200),(102,199), \cdots$, $(150,151)$. In this case, the number of guests in each pair of rooms should be greater than the capacity of the larger room in the pair, otherwise, the hotel manager could arrange the guests in the pair of rooms to the larger room. Thus, the total number of guests in these room pairs should be at least $201+200+\cdots+152=8825$. This indicates that when $n \leqslant 8824$, the hotel manager can arrange the guests as described.

Below is an example to show that when $n \geqslant 8825$, the hotel manager may not be able to arrange the guests as described.

Arrange the rooms in increasing order of their capacity. Assume that in the first 50 rooms, there are 76 guests each, and for $151 \leqslant k \leqslant 200$, in the room with capacity $k$, there are $k-75$ guests. Then the total number of guests currently in the hotel is $76 \times 50+(76+77+\cdots+125)=8825$.

Consider any two rooms with capacities $a$ and $b (a < b)$. If the total number of guests in these two rooms is greater than $b$, then in the constructed example, the extra guests can be sequentially arranged in any room that is not yet full.