On the website, the explanation that a knight's tour on a $6\times6$ board is possible is the continued proof of around $1\frac{1}{2}$ pages! It will be great if one of you could provide a simple, concise proof that the $6\times6$ is possible.
After being asked this question I tried to formulate conjectures to try and prove that the 6x6 is possible by finding solutions based on rules that I thought might work, such as:
However, I still couldn't implement this otherwise-logical 'theorem' in the 6x6 -- I kept running into problems!
Even when I did get it, the vague explanation is just unsatisfactory, and I couldn't really define my method.
I still can't find a way to prove that it really is possible without just giving a straight-up solution.