On twelve of the thirteen black discs are placed numbered counters or grasshoppers. 

Can you reverse their order, so that they shall read, 1, 2, 3, 4, etc., in the opposite direction, with the vacant disc left in the same position as at present?

Move one at a time in any order, either to the adjoining vacant disc or by jumping over one grasshopper, like the moves in draughts. The moves or leaps may be made in either direction that is at any time possible. 

What are the fewest possible moves in which it can be done?

See answer

Math Genius