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© Copyright 1996, Jim Loy
Hey, I'm already famous. It is just that nobody knows it. That's a
joke. I may someday become famous, but not for solving this puzzle. Diagram #3
(reproduced on the left) is a scam. It is not a solution. And you have earned a
virtual pat-on-the-back, if you were skeptical of my claim of having solved it.
You get an additional virtual pat for discovering the flaw in diagram #3.
FLAW: The flaw in my proposed "solution" is that
two lines go from B to X, and two lines go from C to Z. On the right, I have
colored the paths to make this clear.
PROOF: If you want to see the proof that the puzzle is impossible, here goes:
In our path
puzzle, we may draw a path directly from A to X. Or we may take a more indirect
route. We see both ideas in this diagram. These two paths, which look so
different, are essentially the same identical path. So we can simplify our
diagrams considerably. The false "solution" which I proposed, above,
can be unwound to show that it is obviously not a solution.
Well, in our
puzzle, we can easily connect A and B to the three points X, Y, and Z. In this
diagram, B is shown north of the three points. It doesn't matter where B is, so
I moved it to simplify the diagram.
Well, our six paths have "partitioned" the plane into three regions:
This is true, whether the lines are straight, or wrap around our points. We always get three regions.
This exterior can be viewed as just another interior, defined by those points. Pretend that these points are on a globe. A and B are at the two poles, and X, Y, and Z are on the equator, and our paths are lines of longitude. We have partitioned the surface of our globe into three equivalent regions.
Well, our sixth point C will be in one of these three regions. And it is now easy to see that no matter which region C is in, there is one point (X, Y, or Z) that you can't draw a path to (the one point that is not in that region).
So the problem is impossible.
This problem is part of Graph Theory. Graph Theory benefits from some of the ideas of Topology. Topology is sometimes called "rubber sheet geometry." This reflects the fact that certain things do not matter in a Topology problem. Distances, specific positions, and certain orientations do not matter. The plane can be stretched like a rubber (or modeling clay) sheet.
The above puzzle is a simplified version of a problem that electronic circuit board designers face. If all connections on a circuit board are limited to one side of a board, many electronic circuits are impossible. But if you are allowed to go into a third dimension (with wires or by connecting to the second side of the board) then all electronic circuit paths are possible.
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