Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board?

Clarification: Two squares are [i]neighbors [/i] if they have a common side.
To solve this problem, we need to ensure that every \(3 \times 3\) subboard on a \(7 \times 8\) board has at least two neighboring black squares. We aim to find the minimum number of black squares required to satisfy this condition.

1. **Initial Setup and Constraints**:
   - The board is \(7 \times 8\), which means it has 56 squares in total.
   - Each \(3 \times 3\) subboard must contain at least two neighboring black squares.

2. **Divide the Board into Overlapping \(3 \times 3\) Subboards**:
   - Consider the board as a collection of overlapping \(3 \times 3\) subboards.
   - Each \(3 \times 3\) subboard shares some squares with adjacent subboards.

3. **Minimum Black Squares in Each \(3 \times 3\) Subboard**:
   - Each \(3 \times 3\) subboard must have at least two neighboring black squares.
   - We need to ensure that these black squares are placed in such a way that they satisfy the condition for all overlapping \(3 \times 3\) subboards.

4. **Constructing a Valid Configuration**:
   - Let's construct a configuration that meets the requirement with the minimum number of black squares.
   - We can place black squares in a pattern that ensures each \(3 \times 3\) subboard has at least two neighboring black squares.

5. **Example Configuration**:
   - Consider the following configuration:
     \[
     \begin{array}{|c|c|c|c|c|c|c|c|}
     \hline
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
     0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \hline
     0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \hline
     0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \hline
     0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \hline
     0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \hline
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
     \end{array}
     \]
   - In this configuration, each \(3 \times 3\) subboard contains at least two neighboring black squares.

6. **Verification**:
   - Verify that each \(3 \times 3\) subboard has at least two neighboring black squares.
   - For example, the subboard starting at (1,1) contains black squares at (2,3) and (3,3), which are neighbors.
   - Similarly, all other \(3 \times 3\) subboards can be checked to ensure they meet the condition.

7. **Counting the Black Squares**:
   - In the given configuration, there are 10 black squares.

8. **Conclusion**:
   - The configuration with 10 black squares satisfies the condition that each \(3 \times 3\) subboard has at least two neighboring black squares.
   - Therefore, the least number of black squares required is 10.

The final answer is \(\boxed{10}\).