Math Forum / Mathematics / Recreational Math / August 2004

Thinking now after giving the derivation to my prime counting function
I think I have a much better picture of what's been going on, and how
I've been blocked as an amateur mathematicians with my own major
discoveries by others and by myself.

I am a frequenter of these puzzle/math/etc. newsgroups and regularly engage
in posted problems (at least privately).  However, I find little material
for my children.  Does anyone know if there are sites with problems of a
slower speed?  I think it would be great to bend and stretch ...

Say I have (2 - X)^20 = 2^20 - 20 x 2^19 x X + 190 x 2^18 x X^2 - 1140
x 2^17 x X^3...
later on in the equation
= 2^20 - 5x 2^2 x 2^19 x X + 95 x 2 x 2^18

There's a classic example of how inappropriate odds for opposite
outcomes can allow the astute better to always make a profit. Using the
upcoming election as an example, suppose Cheney believes that Bush's
odds of winning the election are 3/4. Also suppose that Edwards believes

What is pascal's triangle commonly used for; statistics, physic's
equations etc. Say I had (1-2x)^6.
Is there a way of working out pascal's triangle without physically
writing it out?

Imagine a line of consisting of the n-th powers
of successive integers, extending indefinitely in
both directions.  Call this line 0.  Thus,
line 0:  ..., (-1)^n, 0^n, 1^n, 2^n, 3^n, ...

I'm trying to solve something that is being explained to me in a text
book.
The factorial is this
6! x 4!

Here's a little something I've not seen before. If a^2+b^2=c^2 and
u^2+v^2=w^2 are primitive Pythagorean triangles, where a and u are
primes, and
v > b, then

This may seem trivial, but I've been thinking (working) on this problem for
a while.
The problem is "Write the equation for the line describing the position the
point (end of the chord) of a pendulum as it moves parallel to a plane.  The

Twin primes are just primes that differ from each other by 2, like 5
and 7.
My prime counting function is
dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,

I'm kind of getting back into mathematics, and have been looking at the
various features of some of the handheld calculators, which seem pretty
capable in their own right.  What kind of confuses me is that given that
w/ a graphing calculator you are getting a highly portable,

Back in my graduate school days, I used to teach introductory calculus. One
of the standard homework questions was to find the following integral:
   Integral dx / ( x + a )    where a is a constant
Occasionally, I would see students solve the problem as follows:

On the magic newsgroup, I was discussed the possible uses of dice loaded
with neodymium magnets. Imagine you have two dice called N1 and N2. When
you put a NORTH magnet under the table, the dice come up '1' and '2'. A
SOUTH magnet would give you '6' and '5'.

Proginoskes (Christopher Heckman),
In an unrelated message, you mentioned,
  Occasionally, there are results by the "gifted high school student"; for
  instance, the result that there exists a real number u such that

Here is a problem by J.A.H. Hunter.
"Good morning, and how are you today?" greeted Mrs. Jones, meeting
her neighbor in the store. "Not too well," replied Mrs. Brown,
stopping to put some of her many parcels down on the counter. "If you