Answer to Question #334 in Combinatorics | Number Theory for Gregory
Is it possible to set 10 bus routes in the city and set the stations on them so that if we take any 8 of them there always will be a station not belonging to any of them, but any 9 routes pass through all the stations?
Yes. Hold on 10 pairs of intersecting lines. Let the routes pass through these lines and the stations are the points of their intersection. Any 9 routes pass through all the stations, because every station on the remaining line also belongs to one of the 9 lines corresponding to these routes. Any 8 routes don’t pass through the station on the intersection of two remaining lines.