This puzzle is part of the Monthly Topic Challenge #7: Board games.

Place a knight on each of the white squares on the 8x8 chessboard. That would make a total of 32 knights.
Is it possible to use a finite number of moves to achieve the goal of moving all the knights onto black squares? If yes, what is the smallest number of moves?

Bonus question:
If we're using knights in distinct colors, is it possible to use a finite number of moves to achieve the goal of turning the original arrangement of knights into its mirror image (along its orthogonal axis)?

Note: Capturing is not allowed here.

Solve the task described above.

(optional) What if we're using queens or rooks?

  • 1
    $\begingroup$ The first question has a trivial answer. The distinct-colors bonus is more interesting. $\endgroup$ Feb 3 at 3:06

2 Answers 2


The first question is quite easy.

The knights within a 2x4 board can swap square colours, so just split the 8x8 board up into 2x4 rectangles like this:
Part 1 solution
Note that this must be optimal as each knight has only done 1 move, and each knight must make at least 1 move to land on the other colour.

Mirroring the board is a lot more tricky.

The move sequence shown here swaps two pairs of knights on two adjacent rows. You can apply this twice two swap all 8 knights on two adjacent rows. If you then apply the first solution, the board has become mirrored. Part 2 solution
This probably is not optimal, but thankfully the question did not ask that.

The optional question is pretty easy again

Having them all move to another colour square is trivial and takes 32 moves just like the knights.
To mirror the arrangement, have all the rooks/queens in a 2x8 rectangle walk around in a loop until they all have moved 8 squares along to the opposite side. This essentially rotates the 2x8 rectangle 180 degrees. Then have all the rooks/queens move vertically to the other row, which essentially mirrors vertically. The net effect is a horizontal mirroring. You can apply this 4 times to mirror an 8x8 board.


I looked at the optional part with rooks.

There is so much freedom to move around that the possibility to swap goes without saying. The interesting part is to find the optimal way for the mirror swap case.

The number of moves required to swap rooks positions left and right is ...


And here is how

You split the board in 4 horizontal stripes of 2 rows.
Each row can be swapped in the following way.

enter image description here

The number of moves for each piece is given at the bottom, for a total of 20 moves.

And here is why it is optimal

Each row has 4 pieces to swap left and right. Since they cannot move thru each other you need to move at least 3 of them out of the row and back. So one piece can do 1 move, all others must do at least 3. That is a minimum of 10 moves per row.

And for the queens:

The same minimum does not apply. You can move a queen out of the row and in in 2 moves only. That sets a lower bound to 56.

With the same moves as above, you save 2 moves with D and E going diagonally. This reduces the count to 18 per stripe or 72 for the board. I don't think this is optimal.

  • $\begingroup$ As for the knights, the lower bound of 112 moves is achievable. $\endgroup$ Feb 4 at 22:27

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