10. At each of the vertices of a cube sits a Bunchkin. Two Bunchkins are said to be adjacent if and only if they sit at either end of one of the cube's edges. Each Bunchkin is either a 'truther', who always tells the truth, or a 'liar', who always lies. All eight Bunchkins say 'I am adjacent to exactly two liars'. What is the maximum number of Bunchkins who are telling the truth?
10. 004 Suppose that there is a truther at $A$. There must be two liars and one truther adjacent to $A$. Let us suppose, without loss of generality, that $B$ is a truther and $D$ and $E$ are liars. Since $B$ is a truther and is adjacent to $A$, then $C$ and $F$ are liars. This shows that there cannot be more than 4 truthers.
If we now suppose that $G$ and $H$ are both truthers, then each of the Bunchkins' statements fits the conditions. So 4 is the maximum possible number of Bunchkins.