19 Apple Trees

Date: 06/29/2001 at 15:15:30
From: Johnson Truong
Subject: Brain teaser

I have a question that my teacher couldn't answer. We got the question 
at a math competition.

You have 19 apple trees. How could you place them in 9 rows of 5?

They gave us a clue: get outside of the box, starlight, starbright,
first star you'll see tonight.

Date: 06/29/2001 at 15:49:26
From: Doctor Jaffee
Subject: Re: Brain teaser

Hi Johnson,

I'll give you another hint. I was able to solve the problem by 
starting off with a six-pointed star (constructed with two equilateral 
triangles).

See if you can work it from there and if you want to check your 
solution with me, write back. If you are having difficulties, let me 
know and show me what you have done so far, and I'll try to help you 
some more.

Good luck.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/30/2001 at 17:47:11
From: Johnson Truong
Subject: Brain teaser

I tried that but I got only 7 rows.

Date: 07/01/2001 at 13:39:14
From: Doctor Jaffee
Subject: Re: Brain teaser

Hi Johnson,

Your star should look something like this:

                            A
                           X
                          / \
                        //   \\
                       /       \
         F\---------------------------------/B
           \        //            \        /
            \      /               \\    //
             \   //                  \\//
              \ /                     /\
               X                    //  \\
             // \                  /      \
            /    \\              //        \\
        E /--------\------------/------------ C
                    \         //
                     \      //
                      \    /
                       \ //
                        \\
                       D

Now connect A to D, then connect B to E, and finally C to F. If you 
drew everything correctly, those three segments should meet at a point 
in the center of the star.  

The 3 line segments you just drew plus the six sides of the star make 
9 lines. There are 5 points of intersection on each line, and the 
total number of intersections is 19.

For more, see the Math Forum's line puzzle at:

  http://mathforum.org/k12/k12puzzles/rosebush.puzzle3.html   

I hope this helps.  Write back if any of my explanation requires 
clarification or if you have any other question

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/