[bonanova]

An old problem asks how many queens may be placed on a standard chess board so that none of them attacks another. It has been asked at least once in this forum.

Let's now permit some hostility. We ask the maximum number of queens that can be placed on a standard chess board such that each queen attacks exactly one other queen. Escalating, what is the maximum number where each queen attacks exactly two others? Three others? Four others?

For each answer, show the placement: with a sketch, codebox or chess notation, where a1 is the lower left, and h8 is the upper right corner.

Have fun, and please use spoilers.

[Guest]

Dang I wish I knew how to load stuff on here, so I'll have to do it in chess notation.

I'll take queens attacking 2 and 3(hopefully)

I believe it is 14.

H3,H5,H8,G3,F8,E1,D8,C1,B1,A2,A4,A6,A7,A8.

I believe it is 16.

H1,H2,H3,H4,H5,H6,H7,H8,A1,A2,A3,A4,A5,A6,A7,A8.

Those were quick solutions, so I'm not 100% on them, but I think they may work out.

[Guest]

10 is the max.

A1, H1, C2, E2, G3, B4, G5, B6, D7, F7.

[Guest]

i'm not sure what the most is, but the fewest is 8.

D3 B ,E3 B, C4 W, F4 W, C5 W, F5 W, D6 B, E6 B.

note that the B and W indicate whether the piece is a black queen or a white queen.

in this position every queen attacks two white queens and two black queens.

can you follow the same pattern with the most?

[Guest]

10 is the maximum:

A1,A7,B4,C4,D8,E8,F3,F5,H2,H6.

There are 8 horizontal lines, 8 vertical lines, 15 NE diagonals & 15 NW diagonals.

Since each queen can only attack 1 other we can group them into pairs.

A diagonal pair will use up 2 horizontal and 2 vertical lines. Therefore only 8 queens can be used just using diagonal pairs.

Horizontal pairs will use up 1 horizontal line and 2 vertical lines.

Vertical pairs will use up 2 horizontal lines and 1 vertical line.

Therefore an optimum solution will use 2 horizontal pairs and 3 vertical pairs or the other way round. This would use 10 queens and would leave 1 empty row or column. (It might also be possible to use 2 horizontal, 2 vertical and a diagonal. I'm not sure.)

[Guest]

Each queen attacking 4 others: 10 Queens can be placed as shown in the fig

Not sure if this is the max possible

Edit: Well, as it turns out, if u upload an image, using a spoiler doesn't prevent "spoiling" (of course it doesn't )... or may be I dont know how to use a spoiler with an image (quite probable, as I didnt read the posting rules) - would be glad to learn!

Edited June 22, 2009 by DeeGee

[Guest]

Got it...

[Guest]

Got it...

I only see 2 of the ten queens being able to attack four other queens. The rest only attack 3 others.

I think it is 20. Just havn't had much time to test it.

[Guest]

I only see 2 of the ten queens being able to attack four other queens. The rest only attack 3 others.

I thought attacking meant in the line of attack whether a there is another queen before or not.. i.e. on a diagonal line of attack if there are 2 queens, both are considered attacked...If you consider only one attacked on a diagonal, then yes, what I did is wrong

[Guest]

Well, going by what James suggested (not counting queens with another one already in front), the number for 4 queens attacking is 24

Hope i got it right this time!

[Guest]

Well, going by what James suggested (not counting queens with another one already in front), the number for 4 queens attacking is 24

Hope i got it right this time!

Sorry, but some of them queens can attack 5 others. When you arrange them, look at each individual queen, and if they were to make a move, look to see how many other queens they can take out.

How do you get that image loaded up on here anyway? You could arrange my example for 2 or 3 if you want(just to see what the OP is asking for). Or the one phillip put up for 1. I don't believe there is a proof for 2,3,or 4. I did it for 2 on a piece of paper. I think the only thing with 2 is it can't be greater than 2n-2(on an nXn board, where in this case n=8)...For 3, I pretty much visualized it in my head, then tested it on paper. For 4 I'm getting tired of scribbling on paper to get what I think is the right answer. Also I'm supposed to be working anyway . Where do you find those chessboard images?

[Guest]

Yeah, I realised that after I had submitted... Well, as for the chessboard images, I made it on excel and saved that part as image (on paint)...

[Guest]

I believe it is 16.

H1,H2,H3,H4,H5,H6,H7,H8,A1,A2,A3,A4,A5,A6,A7,A8.

Not so fast...

I'll give a set up for 18.

H1, H2, H3, H4, H5, H8, G8, F8, E6, D1, C1, C7, B1, B4, A1, A6, A7, A8.

A set up for 21.

H2, H4, H5, H7, G1, G7, G8, F1, F3, E1, D1, C1, C8, B1, B8, A2, A3, A4, A5, A6, A7.