Answer to Question #334 in Combinatorics | Number Theory for Gregory

Question

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?

Answer 1

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.