# Sixteen chess pieces on a square board

It is well known that the eight main chess pieces cannot cover a chess board.

Suppose I have two sets of the eight main pieces. What is the size of the largest chess-like square board all of whose squares can simultaneously be attacked by at least one of the 16 pieces? Two bishops must occupy white cells, and the other two black cells.

• So chess pieces need to be attacked too? Feb 11 at 7:49
• @DmitryKamenetsky Yes, though, as Bass points out, if it is not the case is also an interesting problem. Feb 11 at 17:29
• You can cover a board with 8 pieces if you say a piece covers the square it sits on: fen-to-image.com/image/3K4/8/7Q/3BB3/8/3NN3/R7/1R6
– pkr
Feb 13 at 14:24
• @pkr298 The OP wants the squares occupied by the pieces to also be attacked. Feb 13 at 17:59

I'll get things started with

$$13 \times 13$$: $$\begin{matrix}.&.&.&.&.&.&.&.&.&.&.&R&.\\.&.&.&.&.&.&.&.&.&.&.&.&R\\.&.&.&.&.&.&.&.&.&.&R&.&.\\.&.&B&.&.&Q&.&.&.&.&.&.&.&\\.&K&.&.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&.&.&.&.&K&.&.&.\\.&.&.&B&B&Q&B&.&.&.&.&.&.\\R&.&.&.&.&.&.&.&.&.&.&.&.\\.&.&N&N&.&.&.&.&.&.&.&.&.\\.&.&.&.&N&.&.&.&.&.&.&.&.\\N&.&.&.&.&.&.&.&.&.&.&.&.\\\end{matrix}$$

Improvement:

$$14 \times 14$$: $$\begin{matrix}.&.&.&.&.&.&.&.&.&.&.&.&R&.\\.&.&.&.&.&.&.&.&.&.&.&.&.&R\\.&.&.&.&.&.&.&.&.&.&.&R&.&.\\R&.&.&.&.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&K&.&.&.&.&.&.&.&.\\.&.&.&.&.&N&.&.&.&.&.&.&.&.\\.&.&.&B&.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&.&Q&.&.&.&.&.&.&.\\.&.&.&.&B&.&Q&.&.&.&.&.&.&.\\.&.&.&.&.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&B&.&.&.&K&.&.&.&.\\N&.&.&N&.&B&.&.&.&.&.&.&.&.\\.&.&.&N&.&.&.&.&.&.&.&.&.&.\\.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\end{matrix}$$

(Note: I have treated the question as a classical covering problem, while OP apparently intended that the occupied squares need to be attacked as well. I'm leaving the answer up anyway, since this interpretation yields an interesting puzzle too.)

UPDATE:
Here's the biggest one I got:

15x15

It took surprisingly long to fiddle the placements so that everything fit, so my guess is that the next size is not possible anymore.

Method used:

* Rooks work at optimum efficiency in the corners too, so use them to shrink the board
* The diagonal pieces want to be near the centre of the board to cover the maximum number of squares.
* Use the rest of the pieces to plug any holes left over.
* Try again.
* And again.

Apart from finding suitable software, it's pretty simple to cover a

14 x 14

board with the pieces. Here's how:

Seeing how the queens are suboptimally placed and there's a whole unused piece, this is almost certainly not the maximum. I'll try to cover the next bigger board and update soonish.

• The question implies all squares must be attacked, not just occupied. Feb 11 at 14:37
• @JoelRondeau That's not how these puzzles usually work, so I'm sure OP would have mentioned that unusual requirement.
– Bass
Feb 11 at 16:36
• It does say "all of whose squares" and that was what my answer enforces. Feb 11 at 16:51
• It absolutely does say that. That is not an established chess problem type though, so now we have answers to two problems for the price of one. :-)
– Bass
Feb 11 at 17:14
• What board editor is that? I've only ever found 8x8 editors. Feb 15 at 18:34