8,9 Given a rectangular billiard table with sides 1 and $\sqrt{2}$. A ball is shot from one of its corners at an angle of $45^{\circ}$ to the side. Will it ever fall into a pocket? (The pockets are located at the corners of the billiard table).
At each collision of the ball with the wall, we will reflect the billiard relative to the wall.

## Solution

At each collision of the ball with the wall, we will reflect the billiard relative to this wall. Thus, the ball will always move along a straight line on a "grid paper," where the cells are billiards, and it will never hit a node of the grid (since 1 and $\sqrt{2}$ are incommensurable), i.e., it will never fall into a pocket.

## Answer

No.