Math Forum / Mathematics / Recreational Math / September 2009

Here is a gear problem I solved...
http://mypeoplepc.com/members/jon8338/math/id48.html
... or kinduv.  At least I shed some light on all possible solutions... but
not rotely analytically... but maybe...

I am looking for an efficient prime-counting algorithm to implement in
a functional language.
Some days ago, I found some messages (quite old) from various forums/
usenet groups related to an implementation by Christian Bau. There was

A coin has probability  p  of coming up heads.  Flip this coin  n  
times.  Is there an analytic expression for the expected value of the
length of the longest run?  (A run is a group of consecutive flips all
with the same outcome.)

Find the lengths of the edges of an irregular  triangular pyramid (a
distorted tetrahedron) such that the areas of the faces are 1, 2, 3,
and 4 square units.
John

Have you come up with any interesing math facts that would be a good
addition to www.odd-info.com ?

In the xy Cartersian coordinates, I have two circles with different
radii, r1, and r2.  The centers of the two circles are O1, and O2.
I draw a line that is tangential with both circles. Let's denote the
tangent points as C1 and C2, respectively.

An easyish puzzle and a question of math history.
What is this sum?
  1/3 + 1/6 + 1/10 + 1/15 + 1/21 + ...
Combinatorial proofs only, please!

Full and half parts of a surface are given at :
http://tinypic.com/r/zspmw8/3
How is this surface similar or dissimilar to a cone? Its equation is z
= H ( sqrt ( x^2 + y^2) - LR) / x , H and LR are constants.