The rather perplexing point that the solver has to decide for himself in
attacking this puzzle is whether the shaded numbers (those that are
shown in their right places) are mere dummies or not. Ninety-nine
persons out of a hundred might form the opinion that there can be no
advantage in moving any of them, but if so they would be wrong.
The shortest solution without moving any shaded number is in thirty-two
moves. But the puzzle can be solved in thirty moves. The trick lies in
moving the 6, or the 15, on the second move and replacing it on the
nineteenth move. Here is the solution: 2, 6, 13, 4, 1, 21, 4, 1, 10, 2,
21, 10, 2, 5, 22, 16, 1, 13, 6, 19, 11, 2, 5, 22, 16, 5, 13, 4, 10, 21.
Thirty moves.