Problem 1 · Routes along six paths
Circuits using each path once: village patterns × choices of parallel paths.
Integer answer, at most 4 digitsThree villages are connected pairwise by two paths each, as shown in the picture. Raül wants to walk starting from village $A$, ending back at $A$, and using each of the six paths exactly once. In how many different ways can he do this?
Reasoned solution
Key idea: separate the village itinerary from the choice of the specific path for each leg.
First count village itineraries: closed six-step sequences $A \to \cdots \to A$ crossing each connection ($AB$, $BC$, $CA$) exactly twice. Listing them (e.g. $A\,B\,C\,A\,B\,C\,A$, $A\,B\,C\,A\,C\,B\,A$, $A\,B\,A\,C\,B\,C\,A$, $A\,B\,C\,B\,A\,C\,A$ and their mirror images starting towards $C$) gives exactly $8$.
For a fixed itinerary, each connection is crossed twice and has two parallel paths: we can assign which path is used first in $2$ ways per connection, i.e. $2^{3} = 8$ choices.