16. On the island of Friends and Foes, every citizen is either a Friend (who always tells the truth) or a Foe (who always lies). Seven citizens are sitting in a circle. Each declares "I am sitting between two Foes". How many Friends are there in the circle?
SolUTION
003

If every citizen is a foe then all are telling the truth, which is a contradiction. Therefore, there is at least one friend. Suppose the citizens are labelled ABCDEFG and sat in the same order, with $\mathrm{A}$ (a friend) sat next to $\mathrm{B}$ and $\mathrm{G}$. Then $\mathrm{B}$ and $\mathrm{G}$ are both foes. We now consider $\mathrm{C}$ and $\mathrm{F}$. If $\mathrm{C}$ is a foe, then $\mathrm{D}$ must be a friend (else $\mathrm{C}$ would speak the truth). This makes $\mathrm{E}$ a foe, and $\mathrm{F}$ must be a friend since $\mathrm{F}$ is sitting between two foes, so speaks the truth. In this scenario there are three friends, $\mathrm{A}, \mathrm{D}$ and $\mathrm{F}$.
If $\mathrm{F}$ is a foe, then $\mathrm{E}$ must be a friend (else $\mathrm{F}$ would speak the truth). This makes $\mathrm{D}$ a foe, and $\mathrm{C}$ must be a friend since $\mathrm{C}$ is sitting between two foes, so speaks the truth. In this scenario there are three friends, $\mathrm{A}, \mathrm{C}$ and $\mathrm{E}$.
If $\mathrm{C}$ and $\mathrm{F}$ are both friends then $\mathrm{D}$ and $\mathrm{E}$ must both be foes. In this scenario there are three friends, A, C and F.
Therefore there are three friends.