There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required?

►Consider any one town, it is to be connected with three direct lines with two towns, which are in the same zone and one direct line with each of the other nine towns, which are outside its zone.

►It totally needs 6 + 9 = 15 and there are 12 towns, hence the total direct points to be attached will be 15 × 12 = 180, but every line will be attached to two points.

►Thus the telephone lines needed are 180 / 2 = 90.

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