 a 3x3 board is divided into 1x1 squares/ how many ways are there to color three of the squares so that no two colored squares have a common side?

We are given a \(3 \times 3\) grid and need to determine how many ways we can color exactly three squares such that no two colored squares share a common side. In other words, we want to color three squares that are not adjacent to each other either horizontally, vertically, or diagonally.

### Step-by-Step Solution:

1. **Visualizing the Grid:**
   The \(3 \times 3\) grid has 9 squares, which we can label as follows for clarity:

   \[
   \begin{matrix}
   1 & 2 & 3 \\
   4 & 5 & 6 \\
   7 & 8 & 9
   \end{matrix}
   \]

2. **Choosing Non-Adjacent Squares:**
   To ensure no two colored squares have a common side, they must not be adjacent either horizontally, vertically, or diagonally. Therefore, we are looking for sets of 3 squares such that none of them are adjacent.

3. **Possible Arrangements:**
   We begin by listing combinations of squares and checking which satisfy the condition of no two squares being adjacent.

   - **Corners:** A natural choice is to pick squares from the corners. The corner squares are \(1, 3, 7, 9\).
     - If we color any three of these, none will share an edge. There are \(\binom{4}{3} = 4\) ways to choose 3 squares from these 4 corners.
   
   - **Other Arrangements:** Next, we consider if there are other valid combinations. Trying to choose squares from the center (like square 5) or edges (like squares 2, 4, 6, 8) results in at least one adjacent pair. Hence, they do not satisfy the condition.

4. **Conclusion:**
   The only valid way to select three squares is from the four corner squares. There are \(\binom{4}{3} = 4\) valid ways to do this.

Thus, the number of ways to color three squares so that no two colored squares share a side is \(\boxed{4}\).

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Would you like further details or have any questions about this? Here are some related questions for further exploration:

1. What if we increase the grid size to \(4 \times 4\)? How does the number of ways change?
2. How would the problem change if we were allowed to color four squares instead of three?
3. What if diagonal adjacency was allowed but not horizontal or vertical?
4. How would the solution change if the grid was not square (e.g., a \(3 \times 4\) grid)?
5. Can we generalize the problem to an \(n \times n\) grid?

**Tip:** In combinatorial problems, visualizing the layout and testing small cases can help simplify complex constraints.

a 3x3 board is divided into 1x1 squares/ how many ways are there to color three of the squares so that no two colored squares have a common side?

Learn how to solve a combinatorial problem involving coloring three non-adjacent squares on a 3x3 grid. This detailed solution walks through the steps of grid visualization and using combinatorics, focusing on non-adjacent square selection. Perfect for high school students studying combinatorics.

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