Throne Arrangement

You are tasked with arranging n thrones on an n × n grand ceremonial hall grid. Each throne represents a powerful ruler, and according to ancient kingdom protocols, no two rulers may be able to see each other directly—whether across a row, down a column, or along any diagonal line.

Your goal is to find all distinct valid arrangements of the n thrones on the grid such that no two thrones threaten each other's line of sight.

Each valid arrangement should be represented as an array of n strings, where each string has exactly n characters. In these strings:

• The character 'T' denotes a cell occupied by a throne
• The character '.' denotes an empty cell in the hall

Return all possible valid throne configurations. The solutions may be returned in any order.

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Understanding the Visibility Rules

A throne placed at position (row, col) can "see" (and thus conflicts with) any other throne that lies:

1. In the same row — any position (row, *)
2. In the same column — any position (*, col)
3. On the same diagonal — positions where the absolute difference between rows equals the absolute difference between columns

This means for a valid arrangement, when you place a throne in a particular cell, you must ensure no other throne shares its row, column, or any of its two diagonals.

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Algorithmic Insight

This is a classic constraint satisfaction problem that is elegantly solved using backtracking. The approach involves:

1. Placing thrones row by row
2. For each row, trying every column position
3. Checking if the current position conflicts with any previously placed throne
4. If valid, recursively placing thrones in subsequent rows
5. Backtracking when no valid position exists in a row

The key optimization involves efficiently tracking which columns and diagonals are already "attacked" using sets or boolean arrays, reducing the conflict-checking time from O(n) to O(1).