N-Queens

The n-queens requires placing n queens on an n x n chessboard such that no two queens attack each other. We can approach this problem using backtracking to explore all possible placements.

Here’s an algorithm to solve the n-queens problem:

1. Initialize: Start with an empty chessboard represented as a matrix or a list of strings.

2. Backtrack Function: Create a recursive function that takes the current row and the chessboard configuration.

3. Base Case: If the current row is equal to n, a solution is found. Add the current configuration to the result list.

4. Try Placing Queen: For each column in the current row, try placing a queen there.

5. Check Validity: Verify that the placement is valid (no queens attacking each other).

6. Recursive Call: If the placement is valid, make a recursive call to the next row.

7. Undo Placement: After exploring all possibilities from the current placement, undo it and try the next column.

8. Return Result: Once all possibilities are explored, return the list of solutions.

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class Solution:
    def solveNQueens(self, n: int) -> List[List[str]]:
        def is_valid(board, row, col):
            # Check the same column
            for i in range(row):
                if board[i][col] == 'Q':
                    return False

            # Check upper left diagonal
            for i, j in zip(range(row, -1, -1), range(col, -1, -1)):
                if board[i][j] == 'Q':
                    return False

            # Check upper right diagonal
            for i, j in zip(range(row, -1, -1), range(col, n)):
                if board[i][j] == 'Q':
                    return False

            return True

        def backtrack(row, board):
            if row == n:
                result.append([''.join(row) for row in board])
                return

            for col in range(n):
                if is_valid(board, row, col):
                    board[row][col] = 'Q'
                    backtrack(row + 1, board)
                    board[row][col] = '.'

        result = []
        board = [['.' for _ in range(n)] for _ in range(n)]
        backtrack(0, board)

        return result

This code takes an integer n and returns all distinct solutions to the n-queens puzzle. It uses backtracking to explore all possible solutions, checking for validity at each step.

10 Prerequisite LeetCode Problems

The “N-Queens” involves placing N queens on an NxN chessboard such that no two queens threaten each other. It requires an understanding of backtracking and recursion. Here are 10 problems to build these skills:

1. “Permutations” (LeetCode Problem #46): This problem provides an introduction to recursion and backtracking.

2. “Subsets” (LeetCode Problem #78): This problem further develops the idea of recursion and backtracking to find all possible subsets of a set.

3. “Combination Sum” (LeetCode Problem #39): This problem applies the concept of backtracking to find all possible combinations of numbers that add up to a target.

4. “Generate Parentheses” (LeetCode Problem #22): This problem involves generating all valid parentheses, a concept similar to placing queens on a chessboard.

5. “Letter Combinations of a Phone Number” (LeetCode Problem #17): This problem involves backtracking to generate all possible letter combinations.

6. “Word Search” (LeetCode Problem #79): This problem involves searching for a word in a grid, a problem structure similar to a chessboard.

7. “Sudoku Solver” (LeetCode Problem #37): This problem requires using backtracking to fill a grid, similar to the “N-Queens” problem.

8. “Partition to K Equal Sum Subsets” (LeetCode Problem #698): This problem involves partitioning an array into subsets, which can be viewed as a form of arranging objects (like queens) with certain constraints.

9. “Beautiful Arrangement” (LeetCode Problem #526): This problem is another permutation problem where you need to place numbers in specific positions according to certain rules.

10. “Regular Expression Matching” (LeetCode Problem #10): This problem involves backtracking to find all possible matches to a pattern, which can help with understanding the concept of exploring all possible placements of queens.

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class Solution:
    def solveNQueens(self, n: int) -> List[List[str]]:
        state= [["."] * n for _ in range(n)] 
        res=[]

        visited_cols=set()

        visited_diagonals=set()

        visited_antidiagonals=set()

        def backtrack(r):
            if r==n:                
                res.append(["".join(row) for row in state])
                return

            for c in range(n):
                diff=r-c
                _sum=r+c

                if not (c in visited_cols or diff in visited_diagonals or _sum in visited_antidiagonals):                    
                    visited_cols.add(c)
                    visited_diagonals.add(diff)
                    visited_antidiagonals.add(_sum)
                    state[r][c]='Q' 
                    backtrack(r+1) 

                    visited_cols.remove(c)
                    visited_diagonals.remove(diff)
                    visited_antidiagonals.remove(_sum)
                    state[r][c]='.'                                

        backtrack(0)
        return res

Problem Classification

This problem can be classified as a Board Game/Recreational Mathematics problem based on the problem domain. This classification is because:

• It’s based on the classic chess game, where the placement of pieces (Queens, in this case) follows certain rules.
• The problem involves finding all distinct solutions where N queens can be placed on an NxN chessboard, which falls under the purview of combinatorics - a branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints.
• The nature of the problem involves strategizing on the placement of each queen to prevent any attacks, similar to planning moves in a game.

Other aspects of the problem could also belong to categories such as Constraint Satisfaction Problems (because we’re dealing with certain conditions each queen’s placement must satisfy) or Spatial Problems (since we’re working on a two-dimensional grid). But we focus primarily on the Board Game/Recreational Mathematics aspect, which seems to be the main theme of the problem.

Language Agnostic Coding Drills

This code is a solution to the N-Queens problem using depth-first search (DFS) and backtracking. The N-Queens puzzle is the problem of placing N chess queens on an N×N chessboard so that no two queens threaten each other. Let’s break this down into its smallest units of learning:

1. Lists and Matrices: Understanding of how to create and manipulate lists and matrices.

2. Recursion and Backtracking: Understanding how recursive functions work, and how backtracking can be used to explore all potential solutions to a problem.

3. DFS (Depth-First Search): Understanding how to use DFS to navigate through all possible placements of queens.

4. Sets and Set Operations: Knowledge of how to use sets in a programming language, including adding elements, removing elements, and checking for membership.

5. Understanding Chess Board Layout and Queen’s Movement: Understanding that a queen can move vertically, horizontally, and diagonally on a chessboard.

6. String Manipulation: Basic understanding of how to manipulate strings, for instance how to join a list of characters into a string.

Now let’s order these units in increasing order of difficulty:

1. Lists and Matrices
2. String Manipulation
3. Sets and Set Operations
4. Understanding Chess Board Layout and Queen’s Movement
5. Recursion and Backtracking
6. DFS (Depth-First Search)

The problem-solving approach for the N-Queens problem can be broken down into the following steps:

1. State Representation: Represent the N x N chess board as a list of strings, with ‘Q’ denoting a queen and ‘.’ denoting an empty space.

2. Initialize Data Structures: Initialize the sets that will keep track of the columns, diagonals, and anti-diagonals that are currently being attacked by queens.

3. Backtracking Routine: Implement a function that places a queen on a given row, then recursively tries to place queens on the next rows. If a valid configuration is found, it is added to the result list. If not, the function undoes the move (backtracks) and tries the next column.

4. Validation: Before placing a queen, check whether the current column, diagonal, or anti-diagonal is being attacked by another queen.

5. Traverse: Call the backtracking routine for the first row to start the search for solutions.

6. Return Results: Once all possible placements have been explored, return the list of valid board configurations.

Targeted Drills in Python

1. Lists and Matrices

   Create a list of integers and a 2D list (matrix) of integers.

   1 2	lst = [1, 2, 3, 4, 5] matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

2. String Manipulation

   Create a list of characters and join them into a string using “".join().

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   char_list = ['a', 'b', 'c', 'd', 'e'] string = "".join(char_list) print(string) # Output: 'abcde'

3. Sets and Set Operations

   Create a set, add elements to it, remove an element, and check if an element exists.

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   s = set() s.add(1) s.add(2) print(s) # Output: {1, 2} s.remove(1) print(s) # Output: {2} print(1 in s) # Output: False

4. Understanding Chess Board Layout and Queen’s Movement

   This is a conceptual drill, you can try out the queen’s moves on a physical chessboard or use an online chess platform. The goal is to understand how a queen can move in straight lines both vertically, horizontally, and diagonally.

5. Recursion and Backtracking

   Write a function that generates all permutations of a list of numbers. Use recursion and backtracking.

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   def permute(nums): def backtrack(start): if start == len(nums): result.append(nums[:]) for i in range(start, len(nums)): nums[start], nums[i] = nums[i], nums[start] backtrack(start + 1) nums[start], nums[i] = nums[i], nums[start] result = [] backtrack(0) return result print(permute([1, 2, 3])) # Output: All permutations of [1, 2, 3]

6. DFS (Depth-First Search)

   Implement DFS traversal for a graph.

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   graph = {'A': ['B', 'C'], 'B': ['D', 'E'], 'C': ['F'], 'D': [], 'E': ['F'], 'F': []} visited = set() def dfs(visited, graph, node): if node not in visited: print (node) visited.add(node) for neighbor in graph[node]: dfs(visited, graph, neighbor) dfs(visited, graph, 'A') # Output: DFS traversal starting from 'A'

Specific to this problem, you can also practice creating a 2D list (matrix) of strings, and manipulating the strings within that matrix:

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board = [["."] * 4 for _ in range(4)]
board[1][2] = 'Q'
print(board)  # Output: [['.', '.', '.', '.'], ['.', '.', 'Q', '.'], ['.', '.', '.', '.'], ['.', '.', '.', '.']]