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Connect The Dots (Posted on 2005-02-06) Difficulty: 4 of 5
Take any four points in space. Draw all lines connecting pairs of them. Then draw all lines connecting pairs of points on those lines.

Can the resulting set of points cover all of space?

See The Solution Submitted by David Shin    
Rating: 3.6667 (3 votes)

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re: Smart guys Help | Comment 12 of 15 |
(In reply to Smart guys Help by Hugo)

Hugo,

The point (1,1,1) lies on the line connecting (1/2,1/2,0) and (0,0,-1).  The point (1/2,1/2,0) lies on the line connecting (1,0,0) and (0,1,0), and the point (0,0,-1) lies on the line connecting (0,0,0) and (0,0,1).

As for all (x,y,z) where x, y, and z are greater than 1, try connecting (x,y,z) to the entire line through (1,0,0) and (0,1,0).  That makes some kind of half-plane.  Hopefully you see that that plane intersects the z-axis at some point P.  Going backwards, you can take P, and the point Q at which the line connecting (x,y,z) and P intersects the xy-plane.  P and Q generate the point (x,y,z).  P is found on the line connecting (0,0,0) and (0,0,1), and Q is found on the line connecting (1,0,0) and (0,1,0). 


  Posted by David Shin on 2005-02-08 19:57:26
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