Apple Tree Line Puzzle

Date: 11/07/97 at 11:07:37
From: Doug McCoy
Subject: Line puzzle

Dear Dr. Math,

I'm stuck on a puzzle. A horticulturist planted 13 apple trees in nine
rows with four apple trees in each row. How did he do it? I've tried a
five-pointed star and a six-pointed star and all sorts of odd shapes,
but I'm stumped. Can you help?

Naomi McCoy

Date: 11/20/97 at 08:28:24
From: Doctor Rob
Subject: Re: Line puzzle

Good problem!  It took me and my friends several days to come up 
with the following solution. Credit is due to Dr. Arthur A. Drisko.

Pick three points in the plane not in a line, and call them A, B, 
and C. Draw lines AB and AC. On segment AB pick D, and on AC, pick E.
Draw lines BE and CD, intersecting at F. Draw line DE. Draw line AF,
intersecting DE at G. Draw line CG intersecting EFB at H and ADB at I. 
Draw line DH intersecting AGF at J and AEC at K. Draw line AH, 
intersecting EGD at L and CFD at M.
     

The thirteen points A, B, C, D, E, F, G, H, I, J, K, L, and M are the
places to plant the trees. The nine rows are AIDB, AEKC, AGJF, ALHM,
BFHE, CMFD, CHGI, DGLE, and DJHK, each containing four trees.

Notice the 3-by-3 arrangement GJF, LHM, EKC, and the outside line 
AIBD.

I think this construction makes the following triples of points
collinear: (B,J,L), (B,M,K), (I,J,M), (I,L,K). I have not tried to
prove this, however. This sounds like a theorem from Projective
Geometry.

-Doctors Rob and Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/