First, three moves are sufficent. (Numbering the cups 1 to 7, in any order, you invert cups 1, 2, and 3; then 3, 4, and 5; then 3, 6, and 7.)

Second, the puzzle has no solution if four cups must be inverted at each step (since it is impossible to ever have an odd number of cups the right way up.

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