Question 3
We consider 7 islands $A_{1}, \ldots, A_{7}$. We are allowed to build bridges, either between an island $A_{i}$ and the next island $A_{i+1}$ (for $i \in \{1,2, \ldots, 6\}$), or between an island $A_{i}$ and the last one $A_{7}$ (for $i \in \{1,2, \ldots, 6\}$). In how many ways can these constructions be carried out with as few bridges as possible so that one can travel from any island to any other?
Example for 3 islands instead of 7: the three possible constructions using two bridges are
1) a bridge between $A_{1}$ and $A_{2}$, and a bridge between $A_{1}$ and $A_{3}$
2) a bridge between $A_{1}$ and $A_{2}$, and a bridge between $A_{2}$ and $A_{3}$
3) a bridge between $A_{1}$ and $A_{3}$, and a bridge between $A_{2}$ and $A_{3}$.
Example for 3 islands instead of 7: the three possible constructions using two bridges are
1) a bridge between $A_{1}$ and $A_{2}$, and a bridge between $A_{1}$ and $A_{3}$
2) a bridge between $A_{1}$ and $A_{2}$, and a bridge between $A_{2}$ and $A_{3}$
3) a bridge between $A_{1}$ and $A_{3}$, and a bridge between $A_{2}$ and $A_{3}$.