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What do you mean by 'through' a W? When you add the lines, would it be indefinitely stretching lines, or are their lengths finite?

They will be finite length i.e. line segments.

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W

Though you could theoretically twine one line continuously around the palnet to infinity - all crossing each time, then you will have three dimensional pazzles too

:D But the line segments I mean are finite and do not go beyond the letter "W".

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dacablegy has got the number that I have but in slightly different configuration. Good job. And clearly explained too.

Let see if this is the maximum we get or some one has a better idea.

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