N-Queens: Total Queen Count Patterns

[hariz91]

TL;DR

N=4 has 2 solutions, total queens = 8. Placing solutions on separate boards reveals symmetry. What does this tell us about N-Queens structure?

N-Queens: Total Queen Count Patterns

I've been studying N-Queens solutions and noticed something about the total number of queens across all solutions.

For example, N=4 has 2 solutions. Total queens = 2 × 4 = 8.

This is like preparing 2 separate 4×4 boards and placing one solution pattern on each board. Now we have 8 queens in symmetric positions across both boards.

But if we use only 1 board and switch from the first solution to the second, we lose the first pattern entirely.

From this perspective, what can we observe about the structure or pattern of the N-Queens problem?

Does this suggest that multiple solutions are actually reflections or rotations of a smaller set of fundamental patterns? Or is there something deeper about how queen positions are distributed across all solutions?

I've been looking at the total queen count for various N values:

· N=4: 2 solutions → 8 queens
· N=5: 10 solutions → 50 queens
· N=6: 4 solutions → 24 queens
· N=7: 40 solutions → 280 queens
· N=8: 92 solutions → 736 queens

Does anyone see a pattern here? Or has this been explored before?

 

[jedishrfu]

The number of queens you listed, are they non-attacking queens or the number of arrangements of non-attacking queens?

I believe the number of non-attacking queens is the board size:
- 4 x 4 = 4 queens
- 5 x 5 = 5 queens

 

[hariz91]

The number of queens you listed, are they non-attacking queens or the number of arrangements of non-attacking queens?

I believe the number of non-attacking queens is the board size:
- 4 x 4 = 4 queens
- 5 x 5 = 5 queens

For N=4, each solution has exactly 4 non-attacking queens on the board (as you correctly said, the number of queens equals the board size N).

What I meant by '8 queens' is the total number of queens across all solutions. Since N=4 has 2 solutions, I was imagining placing each solution on a separate 4×4 board. So we have 2 boards × 4 queens per board = 8 queens in total.

I was not suggesting that a single 4×4 board has 8 queens — that would violate the rules. My observation is simply about the total queen count when we look at all solutions together.

 

[pbuk]

I've been studying N-Queens solutions and noticed something about the total number of queens across all solutions.

It is not clear from your post what you have noticed, perhaps you should look at it again and consider whether you have actually observed anything non-trivial at all.

This is like preparing 2 separate 4×4 boards and placing one solution pattern on each board. Now we have 8 queens in symmetric positions across both boards.

Yes, the two solutions for n = 4 are symmetric. We say that there is only one fundamental solution for n = 4. The count of the fundamental and total solutions for n < 28 is on the Wikipedia page.

Does this suggest that multiple solutions are actually reflections or rotations of a smaller set of fundamental patterns?

Yes of course: it is trivial that if S is a solution then rotations and reflections of S are also solutions. This provides a possible 7 additional solutions, however some or all of these transformations may produce the same pattern.

Or is there something deeper about how queen positions are distributed across all solutions?
...
Does anyone see a pattern here? Or has this been explored before?

It has been explored many times before: see for instance the 346 references at https://www.vnster.nl/nqueens/.

There is no closed-form solution for either the total or fundamental solutions so there is no pattern. Multiplying by n doesn't change this.