Derek fills a square $10$ by $10$ grid with $50$ $1$s and $50$ $2$s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N.$ Find the minimum possible value of $N.$
1. **Representation of the Products:**
   - Derek fills a \(10 \times 10\) grid with \(50\) ones and \(50\) twos.
   - The product of the numbers in each row can be represented as \(2^{a_i}\), where \(a_i\) is the number of twos in the \(i\)-th row.
   - Similarly, the product of the numbers in each column can be represented as \(2^{b_j}\), where \(b_j\) is the number of twos in the \(j\)-th column.

2. **Sum of Exponents:**
   - Since there are \(50\) twos in total, the sum of the exponents for the rows is:
     \[
     a_1 + a_2 + \cdots + a_{10} = 50
     \]
   - Similarly, the sum of the exponents for the columns is:
     \[
     b_1 + b_2 + \cdots + b_{10} = 50
     \]

3. **Application of AM-GM Inequality:**
   - By the Arithmetic Mean-Geometric Mean (AM-GM) Inequality, for any non-negative real numbers \(x_1, x_2, \ldots, x_n\), we have:
     \[
     \frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}
     \]
   - Applying this to the exponents \(a_1, a_2, \ldots, a_{10}\):
     \[
     \frac{a_1 + a_2 + \cdots + a_{10}}{10} \geq \sqrt[10]{2^{a_1} \cdot 2^{a_2} \cdot \cdots \cdot 2^{a_{10}}}
     \]
     \[
     \frac{50}{10} \geq \sqrt[10]{2^{50}}
     \]
     \[
     5 \geq 2^5
     \]
     \[
     10 \cdot 2^5 = 320
     \]
   - Similarly, for the columns:
     \[
     10 \cdot 2^5 = 320
     \]

4. **Summing the Minimum Values:**
   - The minimum possible value of \(N\) is the sum of the minimum values for the rows and columns:
     \[
     N \geq 320 + 320 = 640
     \]

5. **Checkerboard Pattern:**
   - This minimum value can be achieved by filling the grid in a checkerboard pattern, such that there are \(5\) instances of \(2\)s in each row and column.

The final answer is \(\boxed{640}\).