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I tried to derive some analytic approximation formula for the lower bound in h-s metric, that is half-turm metric with slice moves for large n. There are 9n possible slice moves for an nxnxn cube, and without using any other relations where would be (9n)^k move sequences of length k. In my simplified model I only used the relations that the 3n slice moves of one axis commute and that there are never two successive moves with the same slice (the latter does not hold in quarter turn metric).