All 92 Solutions to the Eight Queens Puzzle

The Eight Queens Puzzle — placing 8 queens on an 8x8 chessboard so that no two queens attack each other — has exactly 92 valid solutions. This number was first determined by Franz Nauck in 1850, and has since been verified many times over by mathematicians and computer scientists.

To understand these 92 solutions, it helps to know that they are not all truly distinct in a deep mathematical sense. Many are simply rotations or mirror images of each other. When we consider solutions that cannot be transformed into one another through board symmetry as truly different, only 12 fundamental solutions remain.

The relationship 12 → 92 is one of the most elegant aspects of this puzzle. It connects the Eight Queens Problem to group theory, symmetry, and combinatorics — areas of mathematics with deep implications far beyond this puzzle. For the complete mathematical background, see Queens Puzzle Mathematics.

If you want to find these solutions yourself before reading further, the interactive puzzle is a great starting point. The solving guide explains the systematic methods.

This distinction is central to understanding why there are exactly 92 solutions and not some other number.

What is a Fundamental Solution?

A fundamental solution (also called a distinct solution or essentially different solution) is a solution that cannot be obtained from another solution by rotating or reflecting the board. Two solutions are in the same equivalence class if one can be transformed into the other using these symmetry operations.

The 8 Symmetry Operations of a Square

A square (and therefore an 8x8 chessboard) has exactly 8 symmetry operations, forming what mathematicians call the dihedral group D4:

1. Identity: No transformation (leave as is)
2. Rotation 90 degrees: Rotate the board 90 degrees clockwise
3. Rotation 180 degrees: Rotate the board 180 degrees
4. Rotation 270 degrees: Rotate the board 270 degrees clockwise
5. Reflection horizontal: Flip the board left-to-right
6. Reflection vertical: Flip the board top-to-bottom
7. Reflection diagonal (main): Flip along the top-left to bottom-right diagonal
8. Reflection diagonal (anti): Flip along the top-right to bottom-left diagonal

From 12 to 92: The Arithmetic

If every fundamental solution generated 8 distinct variants, we would expect 12 x 8 = 96 solutions. But we get only 92 — a shortfall of 4. This is because one fundamental solution is self-symmetric: it looks the same after a 180 degree rotation. That self-symmetric solution generates only 4 distinct variants (not 8), accounting for the difference: 11 x 8 + 1 x 4 = 88 + 4 = 92.

The self-symmetric fundamental solution is: [5, 3, 8, 4, 7, 1, 6, 2]. Rotating the board 180 degrees transforms this into itself.

Each solution below is written as a sequence of 8 numbers. The number in position i tells you which column the queen in row i is placed in (columns numbered 1–8 left to right, rows numbered 1–8 top to bottom).

For example, [1, 5, 8, 6, 3, 7, 2, 4] means: Row 1 → Column 1, Row 2 → Column 5, Row 3 → Column 8, Row 4 → Column 6, Row 5 → Column 3, Row 6 → Column 7, Row 7 → Column 2, Row 8 → Column 4.

These 12 solutions serve as the "parents" for all 92 solutions. By applying the 8 symmetry operations to each, you can generate the complete set.

Understanding how symmetry transforms solutions is key to seeing the underlying structure of all 92 solutions. Let us walk through how the first fundamental solution generates its family.

Applying Transformations to F1: [1, 5, 8, 6, 3, 7, 2, 4]

Horizontal reflection (mirror left-to-right): Replace each column c with 9−c.
[1,5,8,6,3,7,2,4] → [8,4,1,3,6,2,7,5]

Vertical reflection (mirror top-to-bottom): Reverse the row order.
[1,5,8,6,3,7,2,4] → [4,2,7,3,6,8,5,1] (this is actually F9!)

180 degree rotation: Apply both horizontal and vertical reflections simultaneously.
[1,5,8,6,3,7,2,4] → [5,7,2,6,1,3,8,4]

90 degree clockwise rotation: The transformation is more complex — it requires converting the permutation to a new coordinate system. The queen at row r, column c moves to position (row = c, column = 9−r) in the new orientation.

Each of these 8 transformations produces a (potentially) different valid solution. For F1, all 8 are distinct, giving 8 solutions in the F1 family. For F12 ([5,3,8,4,7,1,6,2]), the 180 degree rotation produces the same arrangement, so only 4 distinct solutions emerge from its 8 transformations.

Solution Family Sizes

This mathematical elegance is why the Eight Queens Puzzle has fascinated mathematicians since its invention. For more on the mathematical connections, visit Queens Puzzle Mathematics.

Below is the complete list of all 92 solutions. Each solution is written as column positions for rows 1–8.

Each number (1–8) represents the column where the queen is placed in that row.

Examining all 92 solutions reveals fascinating structural patterns. These patterns are not merely curiosities — they reflect deep mathematical properties of the puzzle.

Column Frequency in Row 1

How often does the first row queen appear in each column across all 92 solutions?

The distribution is nearly uniform — each column hosts approximately 11–12 first-row queens across all 92 solutions. This reflects the symmetry of the problem: by the board symmetry argument, columns 1 and 8 are equivalent (as are 2 and 7, etc.), so their frequencies should be equal.

Center Columns Are Used More

Across all 92 solutions, columns 4 and 5 appear slightly more frequently in total queen placements compared to columns 1 and 8. This makes intuitive sense: a queen in the center column covers the most diagonal territory, making it harder to place surrounding queens — but when it works, it often leads to balanced configurations.

No Solution Has Adjacent Queens

In every one of the 92 solutions, no two queens occupy squares that are horizontally or vertically adjacent. This is guaranteed by the non-attacking constraint (same row/column rule), but diagonally adjacent queens are also impossible because that would mean the row and column difference are both 1, putting them on the same diagonal.

The Queen Coverage Pattern

In a valid solution, the 8 queens collectively attack (or occupy) a large portion of the board, but some squares are always left unattacked. These "free squares" vary by solution but average around 12–15 per solution.

For more mathematical analysis, see Queens Puzzle Mathematics. To explore solutions interactively, visit the solutions browser.

Reading about solutions is one thing — seeing them play out on a real board is another. The best way to deeply understand why these 92 arrangements work is to explore them interactively.

Our solutions browser lets you cycle through all 92 solutions and see each one visually on an 8x8 board. You can observe which squares are attacked, how queens are distributed, and how symmetry-related solutions compare.

To practice finding solutions yourself using the strategies discussed in this article, visit the interactive puzzle. The puzzle gives you immediate feedback when queens conflict, making it an excellent learning tool.

If you want to understand the deeper mathematics behind these 92 solutions — including connection to combinatorics, group theory, and the N-Queens generalization — see Queens Puzzle Mathematics.