The number of solutions for the N-Queens problem varies significantly with the board size 'N'. For the classic 8-Queens problem, there are 92 unique solutions.
The N-Queens Problem: How Many Solutions for Different Board Sizes?
The N-Queens problem is a classic challenge in combinatorial mathematics and computer science. It involves placing N queens on an N×N chessboard such that no two queens threaten each other. This means no two queens can share the same row, column, or diagonal. An alternate way to think about it is to place N "anythings" on an N×N grid such that none of them share a common row, column, or diagonal.
The Classic 8-Queens Problem (N=8)
For an 8x8 chessboard, a widely studied instance of the problem, there are precisely 92 unique solutions. These 92 solutions represent all possible configurations where eight queens can be placed on an 8x8 board without any threatening another. Interestingly, when considering rotational and reflection symmetries, these 92 solutions condense into 12 distinct patterns or fundamental solutions.
Solutions for Various N Values
The number of solutions for the N-Queens problem grows rapidly and irregularly as N increases. There isn't a simple closed-form formula to calculate the number of solutions for any given N; instead, these numbers are typically found through computational methods, primarily using backtracking algorithms.
The table below outlines the total number of solutions and the number of distinct (fundamental) solutions for various board sizes:
| N (Board Size) | Total Solutions | Distinct Solutions (Symmetry-Reduced) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 0 | 0 |
| 3 | 0 | 0 |
| 4 | 2 | 1 |
| 5 | 10 | 2 |
| 6 | 4 | 1 |
| 7 | 40 | 6 |
| 8 | 92 | 12 |
| 9 | 352 | 46 |
| 10 | 724 | 92 |
| 11 | 2,680 | 341 |
| 12 | 14,200 | 1,787 |
| 13 | 73,712 | 9,233 |
| 14 | 365,596 | 45,752 |
Note: The number of distinct solutions accounts for configurations that are unique even after rotations and reflections are considered equivalent.
For more comprehensive data and solutions for larger N, you can refer to resources like the Online Encyclopedia of Integer Sequences (OEIS A000170) or Wikipedia's N-Queens Problem article.
Finding N-Queens Solutions
Most solutions to the N-Queens problem rely on backtracking algorithms. This method explores possible placements queen by queen. If a queen can be placed in a valid position, the algorithm proceeds to place the next queen. If a queen cannot be placed in any valid spot, the algorithm "backtracks" to the previous queen and tries a different position. This systematic trial-and-error approach guarantees finding all possible solutions, although the computational time increases significantly with larger N.
