Apple Tree Line PuzzleDate: 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/ |
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