Xavier County has a cluster of small towns interconnected by a series of roads, such that there is only a single route between any two towns.
Interestingly, when sorted by distance, the ten routes in the county (measured in km) are unique consecutive primes.
What is the minimal total road length in Xavier? How many towns are there and how are the roads connected in this topology?
A---|-------B
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E --------------------|
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|C
A,B,C,D‚E are the five towns
Routes
AB=5 (1,5+3,5)
AC=11 (1,5+8+1+0,5)
AD=17 (1,5+8+1+6,5)
AE=31 (1,5+8+21,5)
BC=13 (3,5+8+1+0,5)
BD=19 (3,5+8+1+6,5)
BE=33 (3,5+8+21,5)
CD=7 (0,5+6,5)
CE=23 (0,5+1+21,5)
DE=29 (6,5+1+21,5)
The minimal total lenght is the sum of the 4 segments in the diagram 5+9,5+6,5+21,5=42,5 km.
It could be that there is a lower solution, with AB=3 instead of 5.
Edited on December 20, 2024, 4:59 pm
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Posted by armando
on 2024-12-20 16:31:02 |
Solution
