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I really think you should approach this as a physics problem in Dreams rather than a logic one.
The six center pieces would be easy enough to wire up. The tricky bit will be getting the behavior of the other pieces to work in the dreams engine. In a real cube, the 12 edge pieces and 8 corner pieces are held in place by plastic nobs on the inside of the cube that prevent the pieces from popping out, but are not attached to anything.
Try modeling the cube like a real one, making the objects collidable, and see if it works.