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Posted by Anthony Watts
on 24/10/2009, 6:31 am
Some of the visitors to this forum - at least - seem to know their stuff so to speak where mathematics is concerned, so someone might be able to help me with a question I've wanted answering for a long time..
There are 3 basic patterns that would force someone to use 4 colors to fill them in so that no two touching components are the same color..
In other words 4 components can be connected in 3 different ways so that all components are touching all other components..
Namely one component inside three surrounding components, three components within one 'outer ring' and two components inside two 'outer' components..
These examples are obviously 2 dimensional.
I have an idea for a 3 dimensional pattern that may allow for an infinite number of connected components such that every component in the pattern is in contact with all the others, it is this:
Imagine a series of objects that are like spheres with 'rods' attached. The spheres are concentric and each rod passes through all the spheres that are outside of the sphere it is attached to..
My query is this:
Would there be a limit to how many 'spheres with rods attached' could be added to this pattern because there is no longer any room for a rod to be added to the outermost sphere, or could such a pattern be continued indefinitely?
I do not know the solution to this 'puzzle'.
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