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Including but still obstructed by occupied squares, what is the maximum number of squares on a 16x16 board that 8 queens and 8 nightriders can collectively attack at least once each? Even if attacking all 256 squares isn't possible, the maximum number still interests me.

I'm unsure what tags to use for this, or if maybe I should've went to the mathematics SE for this one; please suggest additional tags and let me know if this question would better fit the mathematics SE.

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It's possible to attack all squares.

Q - queen

N - nightrider

Yellow background - attacked only by a queen

Red background - attacked only by a nightrider

Orange background - attacked by pieces of both types

enter image description here

Queens are placed such that 5 longest diagonals in each direction are attacked.

This can be done with just 6 queens, so 2 are redundant and result in queens attacking queens by diagonals. So 4 queens are attacked by other queens, 4 are not. I added nightriders on the edge of the board such that the missing squares are also attacked. Placing nightriders on the edge doesn't block any of the queens' attacks.

One can also see that the nightriders attacking queens could be placed in other ways, so there is more than one solution. An interesting question would be-how many?

edit. now correct...

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    I think I see 2 unattacked squares; the red squares at the upper part of the left edge and lower part of the right edge, the nightriders that seem to supposed to be watching them obstructed by the queens at the furthest corners of the central 8x8 region. However, that queen arrangement looks far superior to what I was previously using; and you're right on point with putting the nightriders along the edges. 254/256 is a massive improvement over 236/256, that's enough to make me believe all 256 squares might be possible; and 254/256 is still really impressive even if 256 is impossible after all.
    – RadiantDarkBlaze
    Nov 9, 2019 at 19:20
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    You're actually right! I'll look into it later if anything can be done about it. Right now im on mobile
    – Sopel
    Nov 9, 2019 at 19:49
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    I thing if you place the knightriders at top middle (one closer to middle) and bottom middle like if they were mirrored horizontally it fizes it
    – Sopel
    Nov 9, 2019 at 19:54
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This is my best attempt thus far, which I believe attacks 236 (all but 20) squares. I used the staircase solution for 8 queens on the central 8x8 region in order to consequentially fully cover the 4 adjacent 8x4 side regions while putting many diagonals through the corner 4x4 regions. Then I placed the nightriders so as to not obstruct any of the queens' lines while also having each nightrider attack 1 unique queen each, by placing them towards the center of the long diagonals still within the corner 4x4 regions.

My personal best attempt at placing 8 queens and 8 nightriders on a 16x16 board so as to attack as many squares as possible

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    I looked at various queen placements and it seems that placing queens such that they don't attack each other produces worse results because it's impossible to cover main diagonals, and a pattern of 4 diagonal squares is not good for knightriders.
    – Sopel
    Nov 9, 2019 at 17:57

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