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Ten Trees Puzzle Summary Page

How to find all six answers to the 10 trees puzzle

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Puzzle: Plant ten trees in five rows with four trees in each row.

There are no tricks. You can actually plant ten trees in a field, one tree per hole, and solve this puzzle. Of course, you're going to solve this puzzle on paper, but you can do it by drawing five lines with four trees (Xs) on each line. This page, and the main collection at the main page will give you all the information you need to find all the solutions.

There are six different "topologically unique" solutions. The solutions you find may look different from any of my examples, but each of your solutions MUST be similar to one of my six solutions. Just imagine drawing your solution onto elastic paper then stretch or shrink one solution to make it similar to one of the others.

Directions: How to find each of the Six Solutions:
Steps 1-5:

Draw five lines.

  • Each line must intersect every other line at different places.
  • No more than two lines may cross at any one point.
  • When you are done drawing your five lines, every line will intersect the other four lines at four different points.
Step 6:

Plant ten trees at the ten places where lines intersect.

Step 7:

Erase all unneeded lines (the line ends are no longer needed). Only the line segments with points at both ends are needed for your solution.

Summary

Working together, we found five of the answers. (click links to see step by step process of how I found each answer.)
  • Our First Answer and Second Answer look like the letter A (though yours may 'topologically' look rather different). You can draw these answers quickly by:
    1. Draw the lines for the capital letter A (your first three lines)
    2. Put a point just above the crossbar of the A (about a third of the way from the crossbar to the top of the A)
    3. Draw your fourth line from the bottom of the left side of the A so that it passes through the point you just added.
    4. Draw your fifth line from the bottom of the right side of the A so that it also passes through that point.
    5. Plant your ten trees (in five rows of four trees in each row)
       
    6. For the "other Letter A" answer, put the point (in step 2) just below the crossbar

      (be sure it is high enough so that the fourth and fifth lines pass through the crossbar before they reach the sides of the letter)
  • Our Third Answer looks like a rocket ship pointed to the sky with fins on both sides. Here's a shortcut to remember how to draw it.
    1. Draw a tall isosceles triangle, but extend the base line out to the left and to the right. It should sort of look like a witch's hat with a brim on both sides.
    2. Put a point in the very middle of the triangle, approximately halfway from top to bottom, and approximately in the middle, left-to-right.
    3. Draw your fourth line from the left end of the base line so that it passes through the point you just put in the middle of the triangle, and continue the line until it touches the right side of the triangle.
    4. Draw your fifth line from the right end of the base line, through the point just added, and to the left side of the triangle.
  • Our Fourth Answer looks like an American Indian Tepee. You can see the opening that lets you crawl into the tent in the middle of the bottom. Here's a shortcut set of directions for drawing the tepee answer.
    1. Draw a wide isosceles triange on your paper. Be sure to leave some white space above the triangle.
    2. Extend the sides of the triangles past the top of the triangle (now you see why I said to leave white space above the triangle).
    3. Plant two trees at both ends of the base line, and put two trees approximately evenly spaced between those outer trees. (In other owords, the four trees divide the base line approximately into thirds.)
    4. Put a point approximately in the middle of the triangle.
    5. Draw your fourth line from the second tree on the baseline up and through the dot you just added, and continue until you meet the extension of the left side of the triangle, up and to the right of the triangle. (sounds confusing, but it looks good).
    6. Draw your fifth line from the third tree on the baseline, through the point, and up to meet the right side of the triangle where it extends outward above and to the left.
  • Our Fifth Answer looks like a steeple on the roof of a house of worship. There is no simple shortcut description for drawing this one.
    1. Draw a right-isosceles triangle with the hypotenuse as the base. Leave as much white space above the triangle as the height of the triangle. (In other words, draw your first triangle on the bottom half of your paper). You may, if you wish, extend the line that forms the left side of the triangle up and to the right into this space above the triangle, but you can do this later if you prefer. (Later, you know where to stop drawing the line).
    2. Bisect (approximate is okay) the hypotenuse and plant a tree at that midpoint.
    3. Bisect (ditto about approximation) the right-half of the hypotenuse and plant a tree at that midpoint.
    4. Imagine that you have bisected the line between the two trees that you just added to the hypotenuse, and put a temporary dot there.
    5. Imagine that you are drawing a perpendicular line straight up from that invisible point and draw it until it is twice the height of the triangle. Put a tree at the top of that invisible line.
    6. Complete the triangle that connects the two trees you added to the base line with the dot you just put directly above and between those points. If the top point is exactly between the two lower points, the new triangle will also be isosceles.
    7. If you have not already lengthened the left side of the triangle, extend that line until it goes through one line and ends at the other line of the new triangle.

     
  • The first four answers can be drawn so that they have an axis of symmetry, so that they look the same in a mirror as they do on paper.
  • The fifth answer has no symmetry.
  • The sixth answer, the one that I'm leaving for you to find on your own, has five different axes of symmetry AND it is rotationally symmetric. In other words, you can see the same image in the mirror five different ways and you can turn one copy of the image and make it exactly congruent with the first image (four other ways).

You can, of course, come up with your own descriptors for each of these diagrams.

The five answers displayed on this page are the answers that many math teachers have never seen. Teachers tend to only know about the easy answer, which is a shape you already know, even if you don't know that you know it.

There is that "sixth" answer, the easy answer. I have not helped you find it. I suspect that you already know that answer. Just in case you still can't find this last answer, you can look back at the work we did for the first three answers.

Here's how to find the sixth answer:

  • Go back to the fourth step of one of the first three answers we found. Click First, Second, or Third, then advance to the fourth step.
  • Here is the fifth step of each of those answers, along with the fifth step of the fourth answer, turned upside down. See if you can figure out where the sixth-answer's horizontal line should go instead of these horizontal lines:
  • With the exception of the horizontal line, all of these images are topologically equivalent to each other. There is only one place that you can put a horizontal line into this fourth-step diagram that is topologically different from these four images.
  • Add one horizontal line to the fourth-step diagram in such a way that you get a new shape.
  • If you topologically stretch the new diagram, vertically and/or horizontally, you should end up with a totally symmetrical shape that you might see at least 50 times in your classroom or in the office of the governor of your state (if you're in the USA that is). If you are outside the USA, you'll have to think of something that would be in our classrooms that would not be in yours.

Using my "draw five lines" technique, you will find many other answers, but TOPOLOGICALLY, they are all going to be identical with one of these six answers, the five below or the one that I hope you found yourself. TOPOLOGY is a fun branch of mathematics that involves drawing shapes on elastic sheets of paper and stretching or twisting them to create new shapes. Topologically, a square and a rectangle are identical because you can stretch one to create the other. If you'd like to know more about topology, click books about topology (at Amazon.com)

Did you find the sixth answer? It's a very common shape that you have seen many times, especially in your classroom and in many churches. In fact, it's the one answer to this puzzle that most math teachers know about. It's the other five answers on this page that many teachers have never seen.