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Problem of the Week 807

Connect the Dots

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This beautiful problem, which has an intuitive appeal that can lead even professional geometers astray, is due to noted mathematician and expositor Victor Klee of the University of Washington.

For a set E in 3-space, let L(E) consist of all points on all lines determined by any two points of E. Thus if V consists of the four vertices of a regular tetrahedron, then L(V) consists of the six edges of the tetrahedron, extended infinitely in both directions.

True or False: Every point of 3-space is in L(L(V))?

I used this problem over four years ago, but because my book containing Macalester PoWs from 1968-1995 will be out soon, this semester is my last chance to reuse some of the best ones that will be included in that collection.

© Copyright 1996 Stan Wagon. Reproduced with permission.

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2 October 1998