# Beautiful geometry: Laser bouncing of walls of a semicircle [closed]

Consider a semicircle with diameter $AB$. A beam of light exits from $A$ at a $58^{\circ}$ to the horizontal $AB$, reflects off the arc $AB$ and continues reflecting off the "walls" of the semicircle until it returns to point $A$.

How many times does the beam of light reflect of the walls of the semicircle (not including when it hits $A$ at the end)?

Note: "walls" of the semicircle refer to the diameter $AB$ unioned with the arc $AB$.

• Are we sure it will ever come back to $A$? – imranfat Mar 14 '16 at 3:56
• Yes the angle is chosen so that it does. – Joshua Benabou Mar 14 '16 at 3:57
• Can the laser bounce of segment $AB$ should it hit the diameter? – imranfat Mar 14 '16 at 3:58
• Yes $AB$ is considered a wall of the semicircle. – Joshua Benabou Mar 14 '16 at 3:58
• Well I find that, if we replace the semicircle with a circle, we get that the beam hits the circle at points which forms 64 degree arcs. – Joshua Benabou Mar 14 '16 at 4:22

After hitting the wall $n$ times the beam is at point $P$ on the circumference with $∠AOP=64n$. The beam first returns to $A$ when $360|64n$ or $n=45$. Thus the beam contacts the arc $AB$ 44 times, because we don't count when it hits $A$ at the end. The beam crosses from the upper half-circle to the lower half-circle or vice verca $64*45/180=16$ times, however this counts includes when the beam hits $A$ at the end, so it actually crosses $15$ times. The total numbner of contacts with the walls is then $44+15=59$.