A geometric gangster shooting
Geometry
Level
3
Ten gangsters are standing on a flat surface, and the distances between them are all distinct. Suddenly, each of them fatally shoots the one among the other nine gangsters that is the nearest. What is the maximum possible number of surviving gangsters?
5
2
8
3
4
6
9
7
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1 solution
See the figure below. All the points are vertices and centers of [almost...all sides distinct] regular pentagons, except that point 3 is slightly tweaked such that distances 1 3 < 1 7 < 3 7 . Then everybody shoots either 1 or 2 , and meanwhile both 1 and 2 shoots 3 , leaving 7 survivors.