Understanding the N-Queens Problem and Diagonal Attack

The N-Queens problem is a classic problem in computer science and chess, where the goal is to place N queens on an NxN chessboard such that no two queens attack each other. In this article, we will explore how to find the diagonal attack of an N-Queen on a given board.

Introduction

The N-Queens problem can be approached using a brute force method, where all possible configurations are generated and checked for safety. However, for larger boards, this approach becomes impractical due to the exponential increase in the number of possibilities.

To solve the problem efficiently, we need to understand how queens attack each other diagonally on a chessboard. The key insight is that two queens can attack each other if they lie on the same diagonal.

Diagonal Attack

Two squares (x1, y1) and (x2, y2) are said to be on the same diagonal if either:

1. x1 + y1 = x2 + y2
2. x1 - y1 = x2 - y2

This means that if a queen is placed at (p, q), it can attack any square (x, y) such that p + q == x + y or p - q == x - y.

Implementing Diagonal Attack in R

To implement this insight in R, we need to modify the existing safe function to check for diagonal attacks. The modified function will take two additional arguments: board and piece.

safe <- function(piece, board) {
  if(sum(board[piece[1],]) < 1 && sum(board[, piece[2]]) < 1) {
    return(TRUE)
  } else {
    for(p = 1;p < length(board);p++) {
      for(q = 1;q < length(board);q++) {
        if(p == piece[1] & q == piece[2]) {
          continue
        }
        if((board[p, q] == 1) && ((p + q == piece[1] + piece[2]) || (p - q == piece[1] - piece[2]))) {
          return(TRUE)
        }
      }
    }
    return(FALSE)
  }
}

This function takes into account both horizontal and vertical attacks, as well as diagonal attacks. It returns TRUE if the queen can attack any square on the board, and FALSE otherwise.

Complete Implementation

To solve the N-Queens problem using this approach, we need to generate all possible configurations of queens on the board and check for safety using the modified safe function.

complete_function <- function(n) {
  chess.board <- matrix(0, nrow = 8, ncol = 8)
  chess.piece <- rbind(rbind(1, 2), rbind(3, 4))
  
  for(p in chess.piece[1,]) {
    for(q in chess.piece[2,]) {
      if(safe(p, q, chess.board)) {
        return("No solution")
      }
    }
  }
  
  for(i = 1; i < length(chess.board);i++) {
    for(j = 1;j < length(chess.board);j++){
      temp <- matrix(0, nrow = 8, ncol = 8)
      
      temp[i, j] <- 1
      
      if(safe(c(i,j), temp)){
        return("No solution")
      }
    }
  }
  
  # Check diagonal attacks
  for(p in chess.piece[1,]) {
    for(q in chess.piece[2,]) {
      board <- chess.board
      for(i = 1;i < length(board);i++){
        if(board[i, q] == 1){
          return("No solution")
        }
      }
      for(j = 1;j < length(board);j++){
        if(board[p, j] == 1){
          return("No solution")
        }
      }
      
      # Diagonal attacks
      for(i in 1:length(board)) {
        for(j in 1:length(board)) {
          if((board[i][j] == 1) && ((i + j == p + q) || (i - j == p - q))) {
            return("No solution")
          }
        }
      }
    }
  }
  
  # If all squares are safe, return "Solution"
  return("Solution")
}

This function generates all possible configurations of queens on the board and checks for safety using the modified safe function. It returns "No solution" if there is no safe configuration, and "Solution" otherwise.

Conclusion

In this article, we have explored how to find the diagonal attack of an N-Queen on a given board. We have implemented a new version of the safe function that checks for diagonal attacks in addition to horizontal and vertical attacks. This allows us to efficiently solve the N-Queens problem using a brute force approach.

References

• N-Queens Problem
• Diagonal Attack

Last modified on 2023-07-29