\section*{Task 4 - 121224}

In a city, a network of at least two bus lines is to be established. This network must satisfy the following conditions:

(1) Each line has exactly three stops.

(2) Each line has exactly one stop in common with every other line.

(3) It is possible to reach any stop from any other stop with one line, without having to change lines in between.

Determine all possibilities for the number of bus lines in such a network.
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In the following, it is assumed that the starting and ending stops of a bus line are added to the three stops that exist according to (1). Each bus line, therefore, runs from the starting stop with exactly one intermediate stop to the end stop. In the following considerations, different letters stand for different stops.

Line 1 serves the three stops A, B, and C. The order here is irrelevant. We write:

Line 1: A-B-C

Line 2, which must exist according to the premise, has exactly one stop in common with Line 1 according to (2). Let this be the stop A for simplicity. D and E are the other two stops of Line 2. Thus:

Line 2: A-D-E

According to (3), there must be Lines 3 and 4 that connect B and D, and B and E, respectively. (A line with the stops B, D, and E cannot exist, as it would share two stops with Line 2.) F and G are the respective third stops of Lines 3 and 4. (F and G cannot be identical, as otherwise Lines 3 and 4 would share two stops, namely B and F = G.). Thus:

Line 3: B-D-F

Line 4: B-E-G

C and D are not yet on a common line. Therefore, there must be another line that serves C and D. Moreover, this must share a stop with Line 4, which can only be G (otherwise B and C or D and E would be on two lines). Thus:

Line 5: C-D-G

Similarly, there must be a line that serves C and E, and this must, due to Line 3, serve the stop F. Thus:

Line 6: C-E-F

Now, a connection between A and F is still missing. This must share a stop with Line 5, which can only be G. Thus:

Line 7: A-F-G

Lines 1 to 7 clearly satisfy conditions (1), (2), and (3). No further line can be added without violating one of the conditions (1), (2), or (3). The only possible number is therefore 7.