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
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