1. ## How Many?

How many different combinations of the position of two black squares in a 3x3 square grid of remaining white squares are possible if you consider any combination that can be rotated to become another combination the same?

To illustrate:

The top set of combinations would be considered the same because the left one, if rotated 90 degrees clockwise, becomes the right. The bottom set of combinations would be considered different because they cannot be rotated in any way to become one another.

2. Are these four coloring equivalent? YES?
How many ways can we choose to color two of the nine sub-squares?
Does each of colorings represent three other colorings? NO!
There is one that represents only one other coloring. Can you tell us which?
So there are basically ${9 \choose 2}$ to color this grid.
But many of those are equivalent.
How do we factor out repetitions? Some happen four times and some happen twice.

3. Hello, Obsidantion!

How many different combinations of the position of two black squares
in a 3x3 square grid are possible if rotations are not counted?

Question: Are reflections also considered equivalent?

Code:
      * - * - * - *       * - * - * - *
| X |   |   |       |   |   | X |
| - * - * - *       * - * - * - *
|   |   | X |       | X |   |   |
| - * - * - *       * - * - * - *
|   |   |   |       |   |   |   |
* - * - * - *       * - * - * - *

Are these considered to be the same?

4. Thanks for the responses.
Good analysis of the puzzle from Plato.

Originally Posted by Soroban
Hello, Obsidantion!

Question: Are reflections also considered equivalent?

Code:

* - * - * - *       * - * - * - *
| X |   |   |       |   |   | X |
| - * - * - *       * - * - * - *
|   |   | X |       | X |   |   |
| - * - * - *       * - * - * - *
|   |   |   |       |   |   |   |
* - * - * - *       * - * - * - *

Are these considered to be the same?
Good point. I should have said that only rotations in which the squares remain aligned with the two dimensional plane of the grid should be accounted for so the two formations in your example wouldn't be considered the same.

5. Hello, Obsidantion!

I was forced to list the cases . . . I found ten.

Place the first square in a corner.

Code:
      *---*---*---*
| X | 1 | 2 |
|---*---*---*
| 3 | 4 | 5 |
|---*---*---*
| - | 6 | 7 |
*---*---*---*
There are 7 positions for the second square.

Place the first square on an edge.

Code:
      *---*---*---*
| - | X | - |
|---*---*---*
| 1 | 2 | - |
|---*---*---*
| - | 3 | - |
*---*---*---*
There are 3 positions for the second square.

Can we generalize this for an $n \times n$ square?
Sorry . . . I have no idea!

6. Originally Posted by Soroban
Hello, Obsidantion!

I was forced to list the cases . . . I found ten.

Place the first square in a corner.
Code:

*---*---*---*
| X | 1 | 2 |
|---*---*---*
| 3 | 4 | 5 |
|---*---*---*
| - | 6 | 7 |
*---*---*---*
There are 7 positions for the second square.

Place the first square on an edge.
Code:
      *---*---*---*
| - | X | - |
|---*---*---*
| 1 | 2 | - |
|---*---*---*
| - | 3 | - |
*---*---*---*
There are 3 positions for the second square.

10 is what I have as well, but if anyone has another answer, feel free to post it and an explanation too. Also post if you have another proof as to why it may be/definitely is 10.

Originally Posted by Soroban

Can we generalize this for an $n \times n$ square?
Sorry . . . I have no idea!
This sounds very hard.