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?
(In reply to
re(2): Detailed Solution by Charlie)
I'll chime in since the problem has already been solved in previous posts.
The fourth point uncovered is (1/2,1/2,1/2). You can find it by performing an affine transformation to map a regular tetrahedron with its four uncovered points onto his tetrahedron.
Or, more simply, take an edge E of his (or any) tetrahedron, translate it so that its midpoint coincides with the midpoint of its skew edge F, and take the endpoints of this translated edge E'. Similarly translate F to get F' and take its endpoints; these four points are uncovered. To see why this transformation is equivalent to the affine one, consider the case with the regular tetrahedron inscribed in a cube.
re(3): Detailed Solution
