Statistical puzzle j8X001

[James L-W]

I've got a real problem, and I hope to tap into the community
brain power.  Credit goes where credit is due:

Supposed I have a 5 X 5 grid of squares, each square 25 cm long.  They
are numbered at random (1-25), and I drop a canister of pick-up sticks
at random over the grid.  I then count the number of times any two
squares share a stick TO THE EXCLUSION OF A THIRD.   I start a matrix:
                         STICK
SQUARE PAIR   1  2  3  4  5  6  7  8  9  10  11  12  13  14...
       #1,#2
       #1,#3
       #1,#4
       #1,#5
       #2,#3

The entries are the value which I will call RELATIVE APPARENT PROXIMITY,
taking uniquely shared sticks as a proxy for distance.  If this value
is summed for all sticks, the value is SIGMA RAP for each taxon pair.

I then send these pairwise scores to a colleague with the challenge to
recreate the correct arrangement of the squares.

My question is: if three squares share a stick, is the value RELATIVE
APPARENT SYNAPOMORPHY for squares i and j independent of the RAP
value for i and k and j and k  *to my collegue*?

HELP!

James