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.