Undergrad Combinatorics - rooks on a chess board

[Lilia]

Doing my combinatorics homework, I just thought that I've made a mistake. When counting the number of ways to place two black and one white rooks on a chess board, I placed the black rooks on black squares and the white one- on a white square? So I chose C(32,1) for the first took. Is that correct? Or it's correct to choose the first square in C(64,1) ways and so on?

 

[FactChecker]

No. A rook of any color can go on a square of any color. The first question to ask is if you have to place them so that they do not threaten each other. A rook can attack any square on the same row or column. The black rooks do not threaten each other.

 

[Lilia]

Yeah I need to place them so those 3 rooks don't threathen each other but that wasn't the question. I chose those in C(32,1)*C(25,1)*C(16,1) ways and there was a conflict if the last one should be C(16,1) or C(18,1) or more probably the sum of the 2. But if I can place a black rook on a white square then I just need to calculate for those numbers, that's makes my work easie

 

[FactChecker]

Be careful to count all unthreatened squares and not to double count any squares.

 

[Lilia]

Okay, I was just confused if I could place a black rook on a white square.

 

[Heinera]

According to the rules of chess, a rook can be on any square no matter the colour.

 

[Lilia]

Yeah I know, it's that my classmates solved the problems not considering that fact and I didn't even notice that, and I just thought about it

 

[Lilia]

Okay so I can't figure out this. Actually I should pick the square of given color, and not the rook. So on 8x8 chess board choose 2 black and 1 white squares and place rooks there so that the rooks don't attack each other. I did this - C(32,1)*C(25,1)*C(16,1) but this is not right. When choosing the 2nd black rook, I can choose it in 2 ways - one, where it crosses lines with the first rook's row's and column's 2 black rooks, and second - with 2 white rooks. So to choose the white one, in the first case there are C(16,1) ways and in the 2nd case - C(18,1). So I wrote C(32,1)*C(25,1)*[C(16,1)+C(18,1)]. Is this correct?

 

[FactChecker]

and second - with 2 white rooks.

I thought there was only one white rook. You need to be very methodical to get these right. Suppose you place the first black rook, then the second black rook. There are two cases -- the second rook is on the same row/column as the first or it is not. Those two cases must be handled separately because they have a different effect on how many squares the white rook can be on.

 

[Lilia]

There are 2 black rooks and 1 white rook. Look at the picture. In the first case the 2nd black rook "closes" 2 white rooks, in the 2nd case it closes two black rooks https://postimg.org/image/qx5pietvf/

 

[PeroK]

I thought there was only one white rook. You need to be very methodical to get these right. Suppose you place the first black rook, then the second black rook. There are two cases -- the second rook is on the same row/column as the first or it is not. Those two cases must be handled separately because they have a different effect on how many squares the white rook can be on.

Or, start with the white rook?

 

[PeroK]

There are 2 black rooks and 1 white rook. Look at the picture. In the first case the 2nd black rook "closes" 2 white rooks, in the 2nd case it closes two black rooks https://postimg.org/image/qx5pietvf/

This is perhaps not a good question for a non chess player. If you put a white rook on the board, can you work out which squares it "attacks"? Perhaps that's what you meant by "closes".

 

[Lilia]

well, actually it closes a black/white square, not a black/white rook

 

[PeroK]

well, actually it closes a black/white square, not a black/white rook

The colours of the squares make no difference in this case (or at all). You could play chess on a board with 64 white squares and it would make no practical difference. It would just be harder on the eye.

 

[Lilia]

the thing is, i should pick the squares, not the rooks, so in one case i had 16 choices for white square, and 18 - in another. but i just chose the white one the first and then the two ones, and this way i get a unique solution

 

[PeroK]

the thing is, i should pick the squares, not the rooks, so in one case i had 16 choices for white square, and 18 - in another. but i just chose the white one the first and then the two ones, and this way i get a unique solution

You've lost me. The white rook can go anywhere. Then, each of the black rooks must be out of the firing line of the white rook. The black rooks don't "threaten" each other, even when they are on the same row. So, it's just about the two black rooks avoiding attack by the white rook.

Does that help?

 

[Lilia]

yeah i know, that's how i found the solution

 

[FactChecker]

Or, start with the white rook?

Good point. That would be simpler.