Question

Given an \(n \times n \times n\) cube containing \(n^{3}\) cells, we are to place \(n\) queens in the cube so that no two queens challenge each other (so that no two queens are in the same row, column, or diagonal). Can the \(n\) -Queens Algorithm (Algorithm 5.1 ) be extended to solve this problem? If so, write the algorithm and implement it on your system to solve problem instances in which \(n=4\) and \(n=8\)

Short answer

Yes, the n-Queens algorithm can be extended to solve the 3D version of the problem. The solution involves modifying the condition checking function to accommodate the third dimension and run the algorithm layer by layer. A successful implementation would print n layers, each an \(n \times n\) matrix, where no two queens threaten each other.

Step-by-step solution

Understand the 2D n-Queens algorithm

First, familiarize with the 2D n-Queens problem. If you recall, the n-Queens problem in two-dimensions asks for all configurations for placing n queens on an \(n \times n\) chessboard such that no two queens threaten each other. A general solution uses backtracking, placing queens column by column (or row by row) and for each column, trying all rows one by one.

Extend the problem to 3D

Now, extend this idea to a 3D cube (an \(n \times n \times n\) cube). This means we need to consider one more dimension. Now we will place queens layer by layer. For each layer, the problem is identical to the 2D case with n layers.

Modify the condition checking function

In 2D, the condition checking function checks whether a queen can be placed at the current cell. In 3D, we need to extend this function to check the additional dimension. Therefore, the function needs to check all layers, rows, columns, and diagonals that the current cell belongs to.

Backtrack if current placement fails

After the condition checking, if we find a cell that can't place a queen (due to it being threatened by another queen), we should remove the queen from the current cell and go back to the previous cell (backtracking). Then, move to the next cell and try to place a queen there again.

Print the solution

Like the 2D version, after all queens are placed successfully, we print the solution. In the 3D case, we'll need to print n layers, each an \(n \times n\) matrix.

Key concepts

Backtracking Algorithm

In solving complex problems like the 3D n-Queens problem, the backtracking algorithm comes in handy. It's an algorithmic approach used for solving constraint satisfaction problems such as word puzzles, Sudoku, and chess problems. Here, we use backtracking to explore potential solutions by trying to build a solution incrementally.

• Start with an empty configuration.
• Place queens one by one. Each time, verify if the current placement is safe.
• If any placement violates the rules, backtrack by removing the last placed queen and try the next possibility.

It's like the trial and error method in other contexts, but here, it works in a structured way, layer by layer, ensuring no overlapping of solutions. Backtracking is particularly effective as it prunes the search space, hence improving efficiency.

3D Chessboard

To solve the extended n-Queens problem, imagine a three-dimensional chessboard, an \( n \times n \times n \) cube. This means there are \( n^3 \) cells in total, forming multiple layers in addition to traditional rows and columns.

• Rows: Run horizontally across each layer.
• Columns: Run vertically across each layer.
• Layers: Formed by the depth of the 3D board, stacked one on top of the other.

In this complex layout, each queen must be positioned in such a way that they do not threaten each other vertically, horizontally, or diagonally across all dimensions.

Algorithm Extension

Extending the n-Queens problem from a 2D board to a 3D board adds a layer of complexity. The algorithm must consider not just placing queens in different rows and columns, but also ensure they don't conflict across layers.
Steps to extend the algorithm:

• Adapt the initial placement logic from 2D to 3D. This involves iterating through each \( n \times n \) layer.
• Modify the checking function to include checks across the third dimension, accounting for the new diagonal possibilities.
• Utilize backtracking to ensure every possibility is explored without any threats; if a placement fails, retry the next configuration.

Queen Placement

Placing a queen on a 3D chessboard requires careful consideration of all three dimensions. Each queen must be placed such that it isn’t attacked by another queen, which involves no shared rows, columns, or diagonals.

• Analyze each cell’s row, column, and layer.
• Ensure no two queens share the same vertical space across any of these dimensions.
• For successful queen placement, check through backtracking to verify the placement is threat-free.

By diligently following these steps, one can ascertain that the queens are placed without threatening each other, ensuring a valid solution.

Diagonal Check

Diagonal checks in a 3D board are more complex because every potential diagonal path must be analyzed. The queen could attack along diagonals that span layers, columns, and rows.

• Diagonals within a single layer: Like in the 2D problem, queens must not share diagonals within each layer.
• Layer-spanning diagonals: Consider movement across layers as another plane for diagonal conflict.
• Both major and minor diagonals need to be assessed across all 3 problem dimensions, requiring rigorous checking.

Each dimension brings new diagonal paths to check, potentially increasing the complexity of each placement.

Layer by Layer Placement

To solve the 3D n-Queens problem, placing queens layer by layer divides the complex problem into more manageable sections. Each layer is treated as a separate board where the traditional 2D problem-solving methods can apply.

• Begin with the first layer, solving it as an independent 2D board.
• Progressively move to the next layer, ensuring the previously solved layers do not threaten the current one.
• Apply the backtracking method for unseen possibilities within and across layers.

This systematic approach helps in managing the complexity of 3D placement by isolating issues to each layer, then synchronizing them holistically across all layers.