Question

In the Reflecting Ball Game a ball can be launched from points 1, 2, 3, 4, 5 or 6 in the direction shown. When a ball hits a side of rectangle $ABCD$ it bounces at a $90^{\circ}$ angle back into the playing field. The path of the ball ends when it hits a corner point A, B, C or D. The path for starting point 5 is shown in the diagram. Each of the 15 non-overlapping squares of the playing field measures 2 cm by 2 cm. What is the length of the longest possible path for a ball launched from a starting point?

Answer 1

Answer:

$24\sqrt{2}$

Step-by-step explanation:

We can measure the number of diagonals each path takes:

From point 1, the path travels 6 diagonals, ending up at point D.

From point 2, the path travels 12 diagonals, ending back at point A.*

From point 3, the path travels 3 diagonals, ending at point B.

From point 4, the path travels 9 diagonals, ending at point C.

From point 6, the path travels 10 diagonals, ending at point D.

Since your question mentions each are 2 cm by 2 cm, this is equivalent to $2\sqrt{2}$.

We can easily conclude that the maximum is 12 diagonals. Thus, Solving our equation gives: 12 * $2\sqrt{2}$   = $24\sqrt{2}$ as our final answer.

*Note: This path will pass through point five, that is, following the diagram you described. If this is true, then there is no need to solve the longest possible path for point 5.

Answer 2

Note that the ball moves only on the diagonals of the 15 non-overlapping squares. Thus we must find the path that traverses the most diagonals.

• From point 1, the path traverses 6 diagonals before ending up at D.

• From point 2, the path traverses 12 diagonals before ending up at A (incidentally, it passes through point 5 in the specified direction so we do not need to check point 5).

• From point 3, the path traverses 3 diagonals before ending up at B.

• From point 4, the path traverses 9 diagonals before ending up at C.

• From point 6, the path traverses 10 diagonals before ending up at D.

Thus the maximum is 12 diagonals, or a length of 12 × 2√2 =24√2 centimeters.