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In total there are $12\times12\times12$ or $12^3$ cubes with edge length $1$cm. Each cube has twelve edges, so the sum of the lengths of the edges on one cube is $12$cm. Therefore the total length of all the edges of all the centimeter cubes is $12^3\times12 = 12^4 = 20,736$cm

This problem is taken from the UKMT Mathematical Challenges.