I'm dealing with some hypercube questions here. The one I'm currently on states:
Find the maximum distance between pairs of vertices in $Q_8$. Give an example of one such pair that achieves this.
I've got the first part of the question down. I drew out $Q_1, Q_2, Q_3, Q_4$, realized that the maximum distance between vertices in $Q_n$ is $n$. After realizing this, it began to make sense to me. $Q_{n+1}$ is created by duplicating $Q_n$ and connecting corresponding vertices. Each vertex from $Q_n$ is connected to exactly one more vertex, and so the maximum distance between vertices will increase by exactly one.
That was the easy part.
How would I possibly show this? I mean. I could spend hours figuring out how to draw $Q_8$, then label all the vertices, and finally present a sequence of eight edges.. But that doesn't sound like the way a mathematician would solve such a problem so it can't possibly be the correct way...
Any help is greatly appreciated!