II. (25 points) Several boxes are unloaded from a cargo ship, with a total weight of 10 tons, and the weight of each box does not exceed 1 ton. To ensure that these boxes can be transported away in one go, the question is: what is the minimum number of trucks with a carrying capacity of 3 tons needed?
II. First, notice that the weight of each box does not exceed 1 ton. Therefore, the weight of boxes that each vehicle can transport at once will not be less than 2 tons. Otherwise, another box can be added.

Let $n$ be the number of vehicles needed, and the weights of the boxes transported by each vehicle be $a_{1}, a_{2}, \cdots, a_{n}$.
Then $2 \leqslant a_{i} \leqslant 3(i=1,2, \cdots, n)$.
Let the total weight of all transported goods be $S$.
Thus, $2 n \leqslant S=a_{1}+a_{2}+\cdots+a_{n} \leqslant 3 n$
$\Rightarrow 2 n \leqslant 10 \leqslant 3 n \Rightarrow \frac{10}{3} \leqslant n \leqslant 5$
$\Rightarrow n=4$ or 5.
Furthermore, it is explained that 4 vehicles are not enough.
Suppose there are 13 boxes, each weighing 13 tons.
Since $\frac{10}{13} \times 4>3$, each vehicle can transport at most 3 boxes. Therefore, 4 vehicles cannot transport all the boxes. Thus, at least 5 vehicles are needed to transport all the boxes at once.