Question 107
Let $K$ be a finite subset of the integers with $k \geq 3$ elements. We say that numbers $a, b \in K$ are connected if and only if there exist numbers $x_{1}, x_{2}, \ldots, x_{k}$ such that:
- $\{x_{1}, x_{2}, \ldots, x_{k}\}=K$,
- $x_{1}=a, x_{k}=b$,
- $|x_{i}-x_{i+1}|$ is an odd number for every $i \in\{1,2, \ldots, k-1\}$.
Prove that the set $K$ contains two numbers that are not connected.
- $\{x_{1}, x_{2}, \ldots, x_{k}\}=K$,
- $x_{1}=a, x_{k}=b$,
- $|x_{i}-x_{i+1}|$ is an odd number for every $i \in\{1,2, \ldots, k-1\}$.
Prove that the set $K$ contains two numbers that are not connected.