Imagine I have a grid of 9 cells. The central cell can not change, and each of the outer cells can be in an on (1) or off (0) state. How many possible unique arrangements of this grid are there, accounting for rotations?

You are describing a 3x3 grid where the central cell remains fixed, and the outer 8 cells can be either on (1) or off (0). The goal is to find how many unique configurations of this grid exist, accounting for the symmetries caused by rotation.

### Step 1: Total possible configurations without rotation
Each of the 8 outer cells can either be on (1) or off (0), giving \(2\) possible states per cell. The total number of possible configurations for the grid, without considering rotational symmetries, is:

\[
2^8 = 256
\]

### Step 2: Account for rotational symmetry
The grid can be rotated by 0°, 90°, 180°, or 270°, which are the symmetries we need to consider. We can use **Burnside's Lemma** (a method from group theory) to calculate the number of unique configurations by averaging the number of configurations that remain unchanged by these rotations.

Let’s evaluate the number of configurations that remain unchanged by each possible rotation:

1. **0° rotation**: All 256 configurations remain unchanged since this is the identity rotation.
   
2. **90° rotation**: For the grid to remain unchanged by a 90° rotation, the values in the outer cells must be arranged symmetrically. This means that all four corner cells must have the same value, and all four edge cells must also be identical. Thus, the grid is determined by two independent choices: one for the corners and one for the edges. So, there are:

   \[
   2 \text{ (choices for the corners)} \times 2 \text{ (choices for the edges)} = 4 \text{ configurations}
   \]

3. **180° rotation**: For the grid to remain unchanged by a 180° rotation, opposite cells must be identical. This divides the grid into 4 pairs of opposite cells, and each pair must be either both on or both off. Thus, there are:

   \[
   2^4 = 16 \text{ configurations}
   \]

4. **270° rotation**: The reasoning for the 270° rotation is the same as for the 90° rotation, yielding 4 configurations.

### Step 3: Apply Burnside's Lemma
According to Burnside's Lemma, the number of unique configurations is the average number of configurations that remain unchanged under each symmetry (rotation). We calculate this as follows:

\[
\text{Unique configurations} = \frac{1}{4} \left( 256 \text{ (0°)} + 4 \text{ (90°)} + 16 \text{ (180°)} + 4 \text{ (270°)} \right)
\]

\[
\text{Unique configurations} = \frac{1}{4} \left( 256 + 4 + 16 + 4 \right) = \frac{1}{4} \times 280 = 70
\]

Thus, the number of unique configurations, accounting for rotations, is **70**.

### Would you like more details on the group theory behind Burnside's Lemma or another aspect of this problem?

Here are some related questions to explore further:
1. How does Burnside's Lemma work for larger grids?
2. What are some other examples of rotational symmetries in mathematics?
3. How would the number of configurations change if reflection symmetries were included?
4. How does group theory apply to combinatorics in general?
5. Can the number of configurations change if the grid is non-square?

**Tip**: Symmetry considerations can drastically reduce the number of distinct configurations in problems involving transformations like rotation or reflection.

Imagine I have a grid of 9 cells. The central cell cannot change, and each of the outer cells can be in an on (1) or off (0) state. How many possible unique arrangements of this grid are there, accounting for rotations?

Discover how to calculate the unique configurations of a 3x3 grid with fixed central cells, where outer cells can be on or off. This solution uses combinatorics, group theory, and Burnside's Lemma to account for rotational symmetries, resulting in 70 unique configurations. Perfect for students exploring advanced symmetry concepts.

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