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I'm no mathematician, but:
You have to ride through every 1k square, no minimal distance for each square.
This means that you just have to visit the intersection of 4 squares to ride (an infinitesimal distance) through each of those 4 squares.
Thus I suspect it simplifies to visiting all the points on a grid of half the size of the original, rounding up.
chez_jay
One for the mathematicians. Imagine a 19x10k made of 1k squares. You have to ride through every 1k square, no minimal distance for each square. How would the theoretical route with the least distance look?