58. The area of a three-pointed star.

Determine the area of the shaded region. (2 Points)

57. Area of a 4 pointed star.

Determine the area of the shaded region. (1 Point)

56. Area of Crescent

Diagonal of the inscribed square is 4 units long. Find area of :

1. Inscribed square
   
2. Circumscribed circle
3. Area of a half-circle on the side of square
4. Area of the arc (of circumscribed circle) on the side of the square
5. Area of the each crescent
6. Area of the four crescents

How does the area of the four crescents compare with the area of the square? (1 point)

55. Calendar Math

Elizabeth and I just figured out how to tell the day of the week given the year month and day. So far we only do dates from 2000 - 2012. Earn four points if you can determine the day of the week ten times in a row without a mistake. (4 Points)

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)