MATH EXPLORERS' CLUB Cornell Department of Mathematics 

Billiards and Puzzles


Two-Hole Billiards

Consider the unusual billiard table shown in the picture below. It contains two special pockets, colored black, one large and one small. There are also eight balls on the table: two green (one small and one large) and six light grey balls of medium size, all placed at fixed positions as shown.

In order to solve this puzzle, one has to send the small green ball into the small black pocket and similarly the large green ball into the large black pocket. However, only two strokes of the cue are allowed. Moreover, one must do this without disturbing any of the six light grey balls, and the green balls must always be in contact with the table. This unusual puzzle appears in the puzzle book of Grabarchuk, "Modern Classic Puzzles", Reference #1.


By simply looking at how the balls are arranged on the board we realize that we need to start with the small green ball, in order not to disturb the grey balls. We also need to remember that a ball reflects from an obstacle at the same angle at which it strikes it. The obstacle can be either the margin (cushion) of the table or another ball. Since we are not allowed to hit the grey balls, the only other ball is the large green one. This leads to a first observation:

Remark: We are allowed to hit either of the green balls.

Notice also that if we hit the small green ball towards the left, there is a large probability that we put it in the large pocket, thus ending the game with a loss. The only reasonable possibility is to hit the cushion somewhere in between ball #1 and ball #2. If we try to hit the cushion somewhere as to put the small green ball in its pocket from this strike only, then we will hit the first gray ball, which is not possible. So we hit the cushion at equal distance between the fist two gray balls. This is shown in the picture below. However, the trajectory of the small green ball must not go on forever, otherwise the fourth gray ball will be disturbed. Hence we assume that there is friction with the table involved and that the player is an expert so that the green ball stops on the line joining the two pockets.

In the second strike we can only hit the small green ball straight into the large green ball. In this way, the small ball will reflect at the same angle, and come back, thus entering the small pocket. Also, the large green ball will move forward directly into the large pocket.

Remark: It is quite difficult (if not impossible) to resolve this puzzle by only striking the two green balls to the cushion, if one is allowed only two strikes of the cue. If there are more strikes allowed, the solution is trivial.

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