Gabriel C: We should not lose the individual aspect of the contest; let's have a separate "chat room" for group problem solving via instant messager (rather than e-mail) and another set of problems for the individual contest.
Alexander G: I am trying to balance between what would work best for you and what is affordable for me. It seems that the present form gives more flexibility. You are free to solve problems individually and e-mail me your solutions. Or, you may try to complete the ideas found by others (but I haven't heard any so far) and discuss them with me in this Mail Room. Or, you may chat with your friends via "instant messages" and even in person (at Modeh Ani or in humanity classes :-) and e-mail your findings to me. Anyway, let's get started and see if further adjustments are needed. So, any ideas?
Danny S: Suppose the diameter of a circle is 1 inch. Then the circumference = pi. Then the radius = 1/2 inch. Draw a hexagon inside the circle so that its corners touch the edge of the circle. Draw 6 lines from the corners of the hexagon to the center of the circle to show the radiuses. The radiuses and the edges of the hexagon will make equilateral triangles. 6 x 1/2 = 3, so the perimeter of the hexagon is 3 inches. The circumference of the circle is longer than the perimeter of the hexagon, so pi > 3.
Alexander G: Excellent! Yet, three comments and a question. First, I am sure, if I ask Danny why the circumference of a circle of diameter 1 is pi, he will say: "This is simply the definition of pi." Next, I am sure that by "a hexagon" Danny meant a regular one. In other words, he divides the circle into 6 equal arcs and connects the nearby division points with straight segments. Furthermore, if I ask why the segments are shorter than the arcs, he'll say "because a straight segment is the shortest path between its ends". Here comes the question: It is true that the segments are 1/2 inch long and that the 6 triangles are equilateral, but why? How do we know that this is true? I am looking forward to hearing further explanations.