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Keep an eye Onto the other (Posted on 2019-07-19) Difficulty: 3 of 5
An odd number of soldiers are stationed in a field in such a way that all the pairwise distances are distinct. Each soldier is told to keep an eye on the nearest other soldier. Show that at least one soldier is not being watched.

No Solution Yet Submitted by Danish Ahmed Khan    
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re: solution | Comment 2 of 4 |
(In reply to solution by xdog)

The proof seems to require the pairing off of all but one of the soldiers.

There are other cases: say the second smallest distance involves a soldier who's one of the two that are involved in the closest distance.

One would like to say that the two soldiers involved in the farthest distance are not being watched, but that's not necessarily true. One would then assume that one of the two is not being watched, but that's also not necessarily true.

Assuming the truth of the statement, I'd think the proof has to be more complicated than that.

  Posted by Charlie on 2019-07-19 10:19:47
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