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13 Point Circle

Date: 8/26/96 at 16:45:43
From: Anonymous
Subject: 5 and 13 point circles

Mr. Fessenden's 8th grade class asks, "Is there an easy way to find 
out how many lines it takes to connect every point on a 13 point 
circle with every other point on the circle? Several students have 
tried to find a pattern or a shape to fit. Some drew lines to the 
center and tried to find a pattern. We would like to know if there is 
an easier way to figure out this puzzle.

Date: 8/30/96 at 16:4:45
From: Doctor Ceeks
Subject: Re: 5 and 13 point circles


We can observe that no line intersects a circle in more than 2 points.  
Therefore, for thirteen points, it is necessary that at least 7 lines 
be used, for 6 lines can only carry at most 12 of the points and there 
will be a thirteenth point left out.

Next, notice that it's always possible to solve the problem using 7 
lines.  One simply pairs up points with lines until 12 points are on 
lines, and then, through the thirteenth point, draw a line not 
parallel to any of the 6 already drawn lines.

This argument works for any odd number of points.

For an even number, say 2M, of points, it is true that you can do it
with M lines.

-Doctor Ceeks,  The Math Forum
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Associated Topics:
Middle School Conic Sections/Circles
Middle School Geometry

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