Frankin 5.

A billiard table has the shape of a $2 \times 1$ rectangle, with pockets located at the corners and at the midpoints of the longer sides. What is the minimum number of balls that need to be placed inside the rectangle so that each pocket lies on a line with some two balls?
Let there be only three balls. A line passing through a pair of balls inside a rectangle intersects the boundary of the rectangle exactly at two points. We have six pockets, so we need at least three lines. Three balls will give three lines only if these lines form a triangle. However, there are only seven lines passing through two pockets, and no three of them form a triangle with vertices inside the billiard table (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_9fc5d17caeb61bf6ddabg-13.jpg?height=337&width=581&top_left_y=0&top_left_x=750)

An example for four balls is indicated in the same figure.

## Answer

Four balls.