20.5. There is a full jug of milk with a capacity of 8 liters and two empty jugs of 5 liters and 3 liters. How to divide the milk into two equal parts
20.5. First solution. If 8-liter, 5-liter, and 3-liter jugs are filled with \(a\), \(b\), and \(c\) liters of milk, respectively, we will denote this as \((a, b, c)\). The following sequence of pourings can be applied: \((8,0,0) \rightarrow (3,5,0) \rightarrow (3,2,3) \rightarrow (6,2,0) \rightarrow (6,0,2) \rightarrow (1,5,2) \rightarrow (1,4,3) \rightarrow (4,4,0)\).

Second solution. If there are three jugs with capacities of \(a\), \(b\), and \(c\) liters, then successive pourings of \(n\) liters of liquid can be conveniently represented in an equilateral triangle with side length \(n\). Each point inside the equilateral triangle with side length \(n\) can be associated with three non-negative numbers \(x\), \(y\), and \(z\) whose sum is \(n\). Specifically, if lines parallel to the sides of the triangle are drawn through a point inside the equilateral triangle with side length \(n\), three smaller triangles with sides \(x\), \(y\), and \(z\) are formed; it is easy to prove that \(x + y + z = n\). Each of the numbers \(x\), \(y\), and \(z\) corresponds to one of the sides of the equilateral triangle. We will consider that the side corresponds to one of the jugs, i.e., we will consider that the jugs with capacities of \(a\), \(b\), and \(c\) liters contain, respectively, \(x\), \(y\), and \(z\) liters. The pourings occur in the region defined by the inequalities \(x \leqslant a\), \(y \leqslant b\), \(z \leqslant c\). Each individual pouring represents a movement from one point on the boundary of this region to another point on the boundary in a direction parallel to one of the sides of the triangle.

Two different variants of the solution are shown in Fig. 20.1.
![](https://cdn.mathpix.com/cropped/2024_05_21_3d5ef740640a71a48194g-250.jpg?height=570&width=1310&top_left_y=728&top_left_x=380)

Fig. 20.1. Pourings