If we do the following instead then we save two whole moves and

consequentally score more all-important

points:

1. 2. 3. 4. 5. 6. 7. 8.
12 31 43 43 32 32 21 21
34 42 21 52 45 51 35 43
56 56 56 61 61 46 46 65

Another useful move to know is how to rotate this:

1 2 3 1 2 3
4 5 6 into this: 4 5 6 .
7 8 9 8 9 7

The most efficient way to do that is as follows:

1. 2. 3. 4. 5. 6. 7.
123 413 413 153 136 136 123
456 526 756 476 457 425 456
789 789 829 829 829 897 897

In the three-by-three game, there are 9! (362,880) different ways of arranging the 9 numbers. All 9! arrangements are reachable.
Of the 362,880 different possible starting positions:

- 1 can be solved in 0 moves (the goal position).
- 8 can be solved in 1 move (by rotating each of the 4 corners either clockwise or anticlockwise).
- 52 can be solved in a minimum of 2 moves.
- 328 can be solved in a minimum of 3 moves.
- 1,996 can be solved in 4 moves.
- 11,336 can be solved in 5 moves.
- 51,582 can be solved in 6 moves.
- 130,042 can be solved in 7 moves.
- 125,929 can be solved in 8 moves.
- 39,706 can be solved in 9 moves.
- 1,880 can be solved in 10 moves.
- 20 can be solved in a minimum of 11 moves.
- 0 can be solved in a minimum of 12 or more moves.

So the 'hardest' possible 3x3 rotation puzzles can be solved in 11 moves. Since there are only 20 of them, I will show them below. Can you find the sequence of 11 moves for any of them?

387 687 789 897 927
654 954 654 654 654
921 321 321 321 381
947 957 967 978 981
852 684 258 654 654
361 321 341 321 327
984 987 987 987 987
657 354 456 564 624
321 621 321 321 351
987 987 987 987 987
645 651 654 654 654
321 324 123 231 312

I will leave the discovery of the 11-move sequences as an exercise for the reader.

Finally, here is how to reverse the entire playfield in 10 moves:

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
987 697 567 574 574 546 546 546 436 413 123
654 584 894 869 826 872 382 132 512 526 456
321 321 321 321 319 319 179 789 789 789 789