Chanukah hexagons

Date: Mon, 12 Dec 94 10:00 EST 
From: Marie Holl
Subject: Re: Ask Dr. Math: on-line math problem solvers

I have an unanswered math problem.  I gave the students the Star
of David for Chanukah.  We tried to find all the triangles,
quadrilaterals, and hexagons in this star.  We were stumped with
the number of hexagons.  Can you help?
__________

Date: Wed, 14 Dec 1994 20:16:40 -0500 (EST)
From: Marie Holl
Subject: Re: Ask Dr. Math: on-line math problem solvers

Dear Dr. Math, 

I'm excited that you might be able to help me.  The Star of David 
that I am working with has the following shape:

                                 /\
                                /  \
                               /    \
                        -------------------
                        \    /\     /\    /
                         \  /  \   /  \  /
                          \/    \ /    \/
                           -------------
                          /\     /\    /\
                         /  \   /  \  /  \
                        /    \ /    \/    \
                       ---------------------
                               \    /
                                \  /
                                 \/

1. How many triangles are there?  We counted 20.

2. How many quadrilaterals are there?  We counted 29.

3. How many hexagons are there? We got lost on this one. We are 
counting irregular hexagons as well.

Thanks for your help. I'm doing this with my 6th grade class.

Marie Holl

Date: 21 Dec 1994 23:56:37 -0500
From: Molly Foster
Subject: Re:  Ask Dr. Math: on-line math problems

Dear Marie:

        Thanks for writing back!  Sorry it took a while for us to get back 
to you.  We've just been finishing up finals here, and most of us are
on vacation right now, but there are still a few of us here answering 
questions!  

        When I counted all the hexagons in the figure you emailed us, I 
got that there were 19.  I'm not guaranteeing there aren't more, but I'll
tell you how I did it, and you can check my work, perhaps.  First, I 
started  with the obvious hexagon--the regular one in between all of the 
points.  So  that is one.  All other hexagons will have to contain two 
sides that form one of the points on the star, right?  So, I focused on all 
the different paths  you might take if you start knowing that the two 
segments that form the point must be in the hexagon.  It's kind of difficult 
to explain exactly how I did it over email, but really all I did was play 
around with the different lines. 

For each point, it seems to me, there are 3 different hexagons that can be 
drawn.  They look like this:                   
                                 /\
                              ---   --
                              \      /
                               \    /
                                \  /
                                 \/

Okay, so that one was a little out of proportion, but do you get the idea?

Here's the next one:
                        /\__
                        \__/

Yeah!  The proportion on that one was pretty good.

And, the final one is:

                /\
                \ \
                /_/

So, for each point there are 3 hexagons.  Thus, there are 18 for all 6 
points. Add this to the one regular hexagon, and you get that there are 
19 hexagons. There may be more that I missed, but the way I did it 
was pretty methodical. 

I hope this helps a little. If you have any other questions, please feel 
free to write back. 

--Sydney, Dr. "math rocks" Foster