356. There are four packages of different masses and accurate balance scales without weights. How can you arrange the packages in order of increasing mass in five weighings?
$\triangle$ Let's number the packages. The masses of the first, second, third, and fourth packages will be denoted by $m_{1}, m_{2}, m_{3}$, and $m_{4}$, respectively.

For the first weighing, we place, for example, the first package on one pan and the second on the other; for the second weighing, the first package on one pan and the third on the other; for the third weighing, the second and third packages on the pans. This allows us to arrange the first three packages in order of increasing mass. Suppose, for example, that the second package is the lightest, and the third is the heaviest of the three (Fig. 41):

$$
m_{2}<m_{1}<m_{3} .
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_0aa0e6ac168e80891953g-110.jpg?height=697&width=257&top_left_y=1556&top_left_x=1436)

Now we weigh the package that is of intermediate mass among these three, in this case, the first, with the fourth. Suppose the fourth package turned out to be heavier.

From the previous, it follows that the heaviest of all four packages is either the third or the fourth. We place these two packages on different pans. For definiteness, let the heavier one turn out to be the third package. Then we get:

$$
m_{2}<m_{1}<m_{4}<m_{3} . \boldsymbol{A}
$$