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:
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.
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.
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.
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.