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sdw
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Joined: January 16th, 2003, 12:29 pm

Another bloody brainteaser . . .

August 15th, 2003, 6:43 am

A finite number of points are drawn on the plane. Each is colouredblack or white. Between ( on the line joining them ) every two blackpoints there is a white point and every two white points there is a blackpoint.Prove that the points must all lie on a straight line - i.e. cannot havethis structure extending into 2 or more dimensions.
 
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Johnny
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Joined: October 18th, 2001, 3:26 pm

Another bloody brainteaser . . .

August 15th, 2003, 7:00 am

Assume a straight line of alternating black and white points.Assume towards a contradiction that there exists a white point not on the straight line.Then there must be a black point between that white point and each white point on the line. So there must be a white point between each of those black points and each black point on the line.So there must be a black point between each of those white points ...... ad infinitum.Each new layer of points implies another layer of points, so by induction there must be uncountable layers of points.Therefore not a finite number of points.
 
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FDAXHunter
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Joined: November 5th, 2002, 4:08 pm

Another bloody brainteaser . . .

August 15th, 2003, 7:06 am

Well, since you said they are drawn on a plane, I can rule out any 3 dimensional structure, unless the plane is topological deformation of the plane, in which case you also need to specify that the straight line must lie in the plane.They must be in a straight line, obviously, as each white can only be connected to blacks via a straight line. If there were a 2-dimensional structure, it would inevitable allow the construction of another straight line, connecting a white one with another white one, or a black one with another black one.