In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$'s needed to be placed to be able to satisfy the conditions
To solve this problem, we need to determine the minimum number of $+1$'s that must be placed in a $4 \times 4$ grid such that if any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. 

1. **Understanding the Condition**:
   The condition states that for any 2 rows and 2 columns deleted, the sum of the remaining 4 numbers must be nonnegative. This implies that in any $2 \times 2$ subgrid formed by the intersection of any 2 rows and 2 columns, the sum of the numbers must be nonnegative.

2. **Maximum Number of $-1$'s**:
   Let's denote the number of $-1$'s in the grid by $k$. We need to find the maximum value of $k$ such that the condition is still satisfied. 

3. **Constructive Example**:
   Consider the following grid configuration:
   \[
   \begin{array}{|c|c|c|c|}
   \hline
   -1 & -1 & +1 & +1 \\
   \hline
   -1 & -1 & +1 & +1 \\
   \hline
   +1 & +1 & +1 & +1 \\
   \hline
   +1 & +1 & +1 & +1 \\
   \hline
   \end{array}
   \]
   In this configuration, there are 6 $-1$'s and 10 $+1$'s. Let's verify if this configuration satisfies the condition.

4. **Verification**:
   - If we delete any 2 rows and 2 columns, we are left with a $2 \times 2$ subgrid.
   - For example, if we delete the first two rows and the first two columns, the remaining subgrid is:
     \[
     \begin{array}{|c|c|}
     \hline
     +1 & +1 \\
     \hline
     +1 & +1 \\
     \hline
     \end{array}
     \]
     The sum of the remaining numbers is $+1 + +1 + +1 + +1 = 4$, which is nonnegative.
   - Similarly, for any other combination of 2 rows and 2 columns deleted, the sum of the remaining numbers will be nonnegative.

5. **Maximality Argument**:
   - Suppose we have more than 6 $-1$'s. Then, there must be at least one $2 \times 2$ subgrid with more than 2 $-1$'s.
   - If a $2 \times 2$ subgrid has 3 or 4 $-1$'s, the sum of the numbers in that subgrid will be negative, violating the condition.
   - Therefore, the maximum number of $-1$'s that can be placed in the grid while satisfying the condition is 6.

6. **Minimum Number of $+1$'s**:
   - Since the grid has 16 cells in total, and we can have at most 6 $-1$'s, the minimum number of $+1$'s needed is:
     \[
     16 - 6 = 10
     \]

The final answer is $\boxed{10}$.