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Puzzling

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    $\begingroup$ To anyone wondering how I came up with this colouring, I tried doing this for 3 cities, then 5 cities, and noticed that reveal spoilerfor n = 2k+1 cities, k+1 R's followed by k B's always seemed to work, and when you start from Rome, you go all the way around. This fixes all the colours of the edges. $\endgroup$ Commented 2 days ago
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    $\begingroup$ I have improved the answer since I posted the above comment. Instead of reveal spoilerk+1 R’s, we need only k R’s. $\endgroup$ Commented 2 days ago
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    $\begingroup$ Correct - well done! $\endgroup$ Commented yesterday
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    $\begingroup$ Nice, I wondered how to prove it was minimal. To rephrase the proof: Rome itself must use an out-and-back route, or else go all the way around and use more than 10 steps. Any out-and-back route consists of an even number of steps, therefore the optimal plan has an even number of steps. But the only way to use an even number of steps starting from only 1 step away is to go 10 steps around the other way - we can't do better than 10 steps. $\endgroup$ Commented yesterday
  • $\begingroup$ @NuclearHoagie. Exactly, I need not have used so many symbols but that's essentially the argument. $\endgroup$ Commented yesterday