Correct solutions were received from Paul Botham, Jimmy Chng Gim Hong, Lou Cairoli, Juan Marivela, Al Zimmermann, Burkart Venzke, Nancy Schwarzkopf, Nick McGrath.

The maximum number of rebounds is five.  Two types of solutions were provided, both interesting.

Consider the triangle in Figure 1 below.  The ball initially makes an angle of a with the edge AC.  Since angle ABC measures 30°, angles BCA and CAB measure 75° each, it follows that b = 105 - a. The ball will return if b ³ 90°, or a £ 15°.  Suppose that a ³ 15°, then g + (75 - a) + (180 - 2b) = 180, and g = 135 - a. The ball will return if g ³ 90°, or a £ 45°.  Suppose that a ³ 45°, then d + b + (180 - 2g) = 180, and d = 165 - a. The ball will return if d ³ 90°, or a £ 75°, which is always the case.  Hence the ball will make at most five rebounds.  

The second solution is very clever and provides more information.  Create six copies of the triangle, all having a common vertex at B, fitting snuggly into a semicircle. See Figure 2.  The angle A'AE measures 15° with successive angles increasing at 15° increments.  Imagine that instead of being "reflected", the ball goes through the edge of the triangle on a straight line path.  The path of the ball is then seen to be a straight line path through the triangles.  The maximum number of rebounds occurs when the initial angle a lies between 60° and 75°.  You can read additional information from this diagram, for instance, when 30° £ a £ 45°, the ball will rebound off BC, then off AB, then off BC again before reaching AC.  

You are visitor number  4479 to this page.