There's an easy proof of this: you can colour the vertices red and blue so that no vertices joined by an edge are the same colour, and there are an odd number of vertices.
So any lap visiting each vertex once has to end on a different colour to the start.
Here's an image showing the colouring property, by David Eppstein. If you start on blue, you have to finish on red, and vice versa.
Just published a new thing to thingiverse: the Seven Triples puzzle
Look upon my works, ye mighty, and despair!
(I'm trying to learn #blender)
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