Counting Ways to Place 8 Rooks on a Chess Board without Attacks

In summary, the problem of placing 8 rooks on an 8x8 chessboard so that no two rooks attack each other has been studied extensively. If the chessboard is oriented, the solution is 8!, or 40,320. However, if the orientation of the chessboard is not known, then the solution is 8+8=16. This is because each rook can be placed in one of 8 rows and one of 8 columns, but the placement of the first rook eliminates one row and one column from consideration for the next rook, and so on. This problem is known as the "rooks on a chessboard problem" and has been the subject of much research.
  • #1
cragar
2,552
3

Homework Statement


How many ways can you place 8 rooks on a chess board so that no 2 rooks attack each other.
Assume the rooks are identical.
Chess board is 8 by 8.

The Attempt at a Solution


To place the first rook I would have 64 choices.
for the second rook I would have 49 choices because I would be limited to a 7 by 7 square.
So the answer should be (64)(49)(36)(25)(16)(9)(4)(1).
I don't think I need to divide by 8! because when I place the rooks that is already taken care of.
 
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  • #2
cragar said:

Homework Statement


How many ways can you place 8 rooks on a chess board so that no 2 rooks attack each other.
Assume the rooks are identical.
Chess board is 8 by 8.

The Attempt at a Solution


To place the first rook I would have 64 choices.
for the second rook I would have 49 choices because I would be limited to a 7 by 7 square.
So the answer should be (64)(49)(36)(25)(16)(9)(4)(1).
I don't think I need to divide by 8! because when I place the rooks that is already taken care of.

I don't think you need to divide by 8! either, assuming the chess board is oriented so you know which side is black and which white. That way each square is uniquely identified and I agree with your solution. However, if you are just presented with a chess board with no orientation, then you would have to account for rotations which you now couldn't distinguish.
 
  • #3
cragar said:

Homework Statement


How many ways can you place 8 rooks on a chess board so that no 2 rooks attack each other.
Assume the rooks are identical.
Chess board is 8 by 8.

The Attempt at a Solution


To place the first rook I would have 64 choices.
for the second rook I would have 49 choices because I would be limited to a 7 by 7 square.
So the answer should be (64)(49)(36)(25)(16)(9)(4)(1).
I don't think I need to divide by 8! because when I place the rooks that is already taken care of.

Interesting problem! :smile:

I get 8+8 = 16. Are you sure you can place all 8 rooks using your method?
 
  • #4
you think the answer is 16? I mean when you place a rook down you eliminate that row and column for your next rook.
 
  • #5
cragar said:
you think the answer is 16? I mean when you place a rook down you eliminate that row and column for your next rook.

Correct. So what relationship do all the rooks have to have to one another in order not to be able to attack each other?
 
  • #6
Each rook has to be in a different row and column, So the first rook has 8 choices for the row, then the next rook has 7 choices for the row ect... it seems like the answer might be 8!. Thanks for your help by the way, I am kinda worried about the region where the places the rooks can attack and how that over laps with rooks not in their row or column
 
  • #7
cragar said:
Each rook has to be in a different row and column, So the first rook has 8 choices for the row, then the next rook has 7 choices for the row ect... it seems like the answer might be 8!. Thanks for your help by the way, I am kinda worried about the region where the places the rooks can attack and how that over laps with rooks not in their row or column

I still get 8+8 = 16. Think geometrically...
 
  • #8
cragar said:
Each rook has to be in a different row and column, So the first rook has 8 choices for the row, then the next rook has 7 choices for the row ect... it seems like the answer might be 8!. Thanks for your help by the way, I am kinda worried about the region where the places the rooks can attack and how that over laps with rooks not in their row or column

It turns out that a lot of study has gone into this problem. Your thinking here for 8! is correct for an oriented chessboard (which is equivalent to dividing your original calculation by 8!). Google rooks on a chessboard for more information than you will want.
 
  • #9
After a PM with LCKurtz, I see now that there are more than the 16 positions that I was thinking of. Thanks! :smile:
 

Related to Counting Ways to Place 8 Rooks on a Chess Board without Attacks

1. How many ways can rooks be placed on a chessboard without attacking each other?

This is known as the n-queens problem, and the answer depends on the size of the chessboard. For a standard 8x8 board, there are 92 possible configurations.

2. How do you approach solving the counting problem with rooks?

One approach is to use combinatorics and permutations to determine the number of possible arrangements. Another approach is to use recursion and backtracking algorithms to find all possible solutions.

3. What is the formula for calculating the number of ways to place n rooks on an n x n chessboard?

The formula is n!, which represents the number of permutations of n objects. This can also be written as n factorial.

4. Can you explain the concept of attacking rooks in this counting problem?

In this context, attacking rooks refers to the situation where two or more rooks are placed on the same row or column on a chessboard. This is not allowed in the counting problem, as the goal is to find all possible configurations where rooks do not attack each other.

5. Is there a generalization of this problem for other chess pieces?

Yes, there are similar counting problems for other chess pieces, such as bishops, knights, and queens. Each piece has its own unique set of rules and constraints that must be considered when solving the problem.

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