[ExI] more math and maybe some fermat

[Will Steinberg]

Probably already a thing (as it always is) but Wikipedia and Wolfram aren't
turning anything up so:

Today I was thinking about squares as the sum of odd numbers, and saw that
the reason was because to create the next square, you must add a row of n
spots and then a row of n-1, summing to 2n-1, as shown:

o o O
*o o O*
OOO

and I saw it was easily extendable to any dimension (e.g. for cubes we add a
face of x*( blocks, then one of x(x-1), and finally one of (x-1)(x-1); for
quartics, x(x)(x) + x(x)(x-1) + (x)(x-1)(x-1) + (x-1)(x-1)(x-1).

So I generalized it to the equation in the attached file.  Neat.

More interesting was this:

You can see with geometric simplicity that a^2+b^2 can equal c^2 for
9+16=25:

oeeeee
oooeee
oooooe
ooooooo

As we can see, the odd numbers mesh together to make a box of 4*6 +1:
(n-1)*(n+1) + 1 = n^2 -1 + 1 = n^2  (n=5).

What's so difficult is we can see a special reason for this occurring with
the squares, but it is much harder to show how it does NOT occur in the
higher powers.  I think this could lead to an elementary proof of Fermat's
Last Theorem.
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