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@Arducius
I cut the 6×10 out of paper and have been unable to make any other rectangles. Reading the Wikipedia page where you got that graphic it turns out there are only 2 solutions to the 3×20 (both of which use the same pieces). At least one piece in the 6×10 (e.g. L piece) is the mirror of the one in the 3×20 which means that it is impossible to use the pieces in the 6×10 to make a 3×20.
So I am pretty sure the only way to do it is to start with the 3×20 pieces, which you have done. Was that coincidence or deliberate?