54. A simpler web.

Problem 53. is tough. You might be able to wrap you head around this one, then, once you see the principle, give 53. another try. A spider and a fly are on two corners of a triangular web. The spider moves to an adjacent corner randomly once a second. The fly stays put. On average how many seconds does it take for the spider to reach the fly? (2 Points)

53. The Bucky Spider

An fly and a blind spider are on opposite corners of a buckyball (Carbon 60, a soccer ball shape). The fly is stationary and the spider moves at random from one corner to another along the edges only, once a second. On average, how many seconds does it take the spider to reach the fly? (5 Points)

Replace "buckyball" with "cube" and the problem is worth (3 points) .

Replace "cube" with "dodecahedra" and the problem is worth (4 points).

Problem 52. Golden Pentagon

52. A regular pentagon is circumscribed by a circle of diameter length 1, see diagram.

• What is the length of line segment AC? (2 points + 1 point bonus without any trig function)
• What is the area of the pentagon? (2 points)