| Suppose there were 7 roads coming out of city C.
This would make 7 angles. The sum of all these angles is 360
degrees, so some angle must be strictly less than 60 degrees.
(Otherwise we would have 7 angles each of 60 degrees or more
and this adds up to more than 360.)
So we have cities A and B with angle ACB being less than 60 degrees.
But, by the shortest distance hypothesis, AB cannot be the
shortest side of triangle ACB. Since the shortest side is opposite
the smallest angle, this implies that angle C cannot be the
smallest angle, so angle B, say is smaller (or equal) than angle C.
Similarly, angle A can't be larger than angle C for then BC would
be larger than BA and so city B should not have built a road to city C.
Hence all three angles of this triangle are strictly less than 60 degrees.
This is a contradiction. (angle sum less than 180)
Thus, our hypothesis is false, and at most 6 roads can come out of city C.
--------------
Note that by the same reasoning, if 6 roads come out of any city,
then they must make 60 degree angles with each other.
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| In an imaginary galaxy, they made a hyperspace bypass between any pair of
habitable planets which were the CLOSEST to each other, and only such
hyperspace bypasses. Determine the maximum number of hyperspace bypasses
for a habitable planet. Please assume a steady-state model of the universe.
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| Hmmm, this is gonna be tricky to explain. The is a branch of math's that
deals with 'measuring distances' in arbitrary spaces. I hit it during
real and complex analysis, but I'm sure you can get there from other
approaches. Here's what it says about your problem...
Definition:
Given a set of 'points' S,
A 'metric' is a function from SxS to Reals > 0 such that
(1) For all X,Y in S, F(X,Y) = 0 if and only if X = Y
(2) For all X,Y in S, F(X,Y) = F(Y,X)
(3) For all X,Y,Z in S, F(X,Y) + F(Y,Z) >= F(X,Z)
Rationale:
These say (1) the only point 0 distance from here is here,
(2) the distance from here to there is the same as that from
there to here,
(3) there aren't any 'wormholes' = going somewhere else before
you go to your destination shouldn't shorten your
journey.
Example:
S = (a, b, c, d, e, ...)
F(X,Y) = 0 if X=Y
= 1 if X<>Y
In this universe, each city has as many roads leaving it as there are
other cities!
Conclusion:
Maybe you'd better restrict the problem to the 'usual' metric in a
Euclidean space.
/Bevin
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| I thought I had so restricted it, by the assumption of a steady-state model
of the universe :).
Perhaps I'm too abstruse, or my physics is forgotten. Please assume Euclidean
three-space, with an R2 metric: distance = sqrt( dx^2 + dy^2 + dz^2 ).
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