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In the minigame PipeMaze, can a player have no chance, with the maze given, to get any money?

This is pipe maze, if you don't know what it is.

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up vote 3 down vote accepted

It will always be possible for the chest to reach any player. If it wasn't, then there would have to be (at least) two entrances which led to one player.

That, however, would result in a logical inconsistency: suppose we have two different paths, A and B, which both lead to the same exit. By necessity, there must be some part of these two paths which is identical (since they end up in the same exit). Let's use A' to designate the part of the path only in A, B' to designate the part of the path only in B, and C to designate the shared part.

All connections are of the or the variety - we can safely assume the connection at which C starts is a , since the other case works under the same principles.

That means the connection where C starts must

A'  B'
╠B' ╠A'
C   C

By definition, these are the only two options, since A' and B' have been distinct up to this point.

This, then, leads to a contradiction: if we enter from above, we will not be able to go to C, since we have to cross over.

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Slightly simpler: if two entrances led to the same exit, we could flip the maze upside-down, and (since the maze works exactly the same in reverse), get one entrance which leads to two exits (and one entrance which leads nowhere). –  BlueRaja - Danny Pflughoeft Sep 15 '12 at 23:00
    
@BlueRaja-DannyPflughoeft: Well sure, if you want to do it the easy way... :) –  Michael Madsen Sep 15 '12 at 23:30
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