Date: Tue, 17 Jan 2006 04:06:56 +0000
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This is a classic puzzle, one that I first found in the James Fixx's
book _Games for the Superintelligent_. His books are excellent.
However, James only listed one answer to this puzzle. There are three
topologically unique answers to this puzzle.
Here are some hints to help you solve this puzzle.
1. You can do this with a normal field or yard, or you can make a chart
on a flat piece of paper, marking separate X's on the paper to
represent each tree.
2. You don't have to do any trickery. Put one tree in one hole in the
ground. Every tree is in a different place in the yard, and you will
be able to walk from place to place. If you are doing this on paper,
every X will be distinct and visible seperately.
3. Every "row" will be a straight line. You could draw a straight line
through four X's with a ruler. Here's one line to get you started:
x----------x----------x----------x
4. Change your viewpoint. You are NOT going to arrange the rows like
you do rows of chairs in your classroom.
5. Five rows times four trees is equal to 20. 5 4 = 20.
6. You are told to use ten trees. 20 10 = 2. Every tree has to be in
two different rows.
7. Can you think of any way to use five lines to draw geometric figures
that have ten vertices (corners and intersections).
There are lots of hints here. Try to figure out the easy answer for
yourself.
Two more hints:
8. If you draw lines connecting all the rows in the traditional answer,
there will be no right angles, no square corners.
9. The popular answer to this puzzle is a geometric figure that you see
in many places around your community and around your school.
As you might guess, I am NOT going to tell you the easy answer. Reread
all the hints that I have given you above, and look around your
classroom. The answer is one that you have seen, and probably drawn,
hundreds of times in your life.
After you either find the easy answer, or give up on finding the
answer, keep reading because I'm going to tell you how to find one of
the two harder answers.
The unpopular answers are ones that most people don't know about. I
will give you one of those "unlikely" answers. If you are searching
for the answer for a classroom assignment, I strongly suggest that you
search for the well-known answer also.
The SECOND answer
If you like a REAL challenge, try to find the other answer. It is, by
all means, the harder answer to find. It is not based on a regular
geometric figure, but instead, is based loosely on a letter of the
Alphabet. That's ALL the hints that I'm going to give you on this
one. Try to figure out the OTHER answer before you scroll down to see
my version of the other answer.
T H E
O T H E R
A N S W E R
The Other Answer
I'll do it twice. Once in "straight text" and a second time with bold
letters and CENTERING.
Here's how to draw the diagram.
1. Print a capital letter A with the crossbar of the A about two-thirds
of the way down the legs.
2. This takes care of five trees. The first is at the top of the A.
The second and third are at the bottom of the left and right legs. The
fourth and fifth are where the crossbar meets the left and right leg.
Here's what you have so far.
1
/ \
/ \
/ \
4-------5
/ \
2 3
3. Put a sixth and seventh tree halfway between the top of the A and
the crossbar. This now takes care of two rows of four trees. There
are four trees on each leg of the A.
1
/ \
6 7
/ \
4-------5
/ \
2 3
4. Draw a straight line to connect points 3 and 6, and another
straight line to connect points 2 and 7. I'll let you draw the actual
lines in the sample. I can't draw lines in text at this angle.
5. Put the remaining three trees where these lines meet.
Here's another copy of this answer without the lines, and with all
points listed.
1
6 7
8
4 9 0 5
2 3
In this diagram, there are five rows: 1642, 1753, 4905, 2987, and
3086. Notice how every tree is in two different rows.
THE THIRD ANSWER
The third answer, also a HARD-TO-FIND one, is loosely related to the
answer that we just found above. I'm not going to tell you how to find
it, but I will tell you this much. Three of the rows and nine of the
trees are in identical places. Move exactly one tree and you will have
the third answer.
Did you find the simpler answer? It's so simple, I'm not going to give
it to you. You'll be glad I didn't give you the answer when you
finally figure it out.
John "Maze Man" Knoderer
John@Mazes.com
THEMAZEMAN@aol.com