The mainland counties of England form a graph with counties as vertices and edges as touching borders. Is there a Hamiltonian path one can take? This is not homework, I just have an idea for a holiday around England where I visit every county only once!
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Let's assume we can access the Isle of Wight through Hampshire. Then the answer is yes:
N.B.: This uses ceremonial counties instead of administrative counties; see comments for discussion.
Edit (after a comment below): The background image from Wikipedia. I found the path mostly by luck, with the knowledge that I had to start in the Isle of Wight and finish in Cornwall.