Hello, Bruce!

We could use a **lot** of clarification.

. . But I'll take a wild guess . . .

First of all, I'll guess that a "triangulated triangle" is something like this

. . (not necessarily equilateral): Code:

*
/ \
* *
/ \ / \
* - * - *
/ \ / \ / \
* - * - * - *
/ \ / \ / \ / \
* - * - * - * - *

We can call this a triangulation of "order 4":

Then each of the __smallest__ triangles is labeled with

One arrangement looks like this: Code:

x
/ \
y z
/ \ / \
z x y
/ \ / \ / \
x - y - z - x
/ \ / \ / \ / \
y - z - x - y - z

Reading around the outside perimeter, the sequence of vertices is:

. .

That is, occurs times.

Further sketching convinced me that this always happens.

For a triangulation of order , the occurs times.

. . Hence, the segment appears times.

If "the triangulation labeled " means the __order__ of the triangulation,

then the number of xy-segment not only has the same parity as the order,

. . it is actually *equal* to the order.

Is this *anything* like what you meant?