A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure:
Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet.
[img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]
1. **Define the dimensions of the original sheet:**
   Let the length of the original rectangular sheet be \( a \) and the width be \( b \).

2. **Determine the dimensions of the rectangles after each fold:**
   - After the first fold, Rectangle 1 has dimensions \( a - b \) and \( b \).
   - After the second fold, Rectangle 2 has dimensions \( a - b \) and \( 2b - a \).
   - After the third fold, Rectangle 3 has dimensions \( 2b - a \) and \( 2a - 3b \).

3. **Calculate the perimeters of the rectangles:**
   - Perimeter of Rectangle 1: \( P_1 = 2(a - b) + 2b = 2a \)
   - Perimeter of Rectangle 2: \( P_2 = 2(a - b) + 2(2b - a) = 2a - 2b + 4b - 2a = 2b \)
   - Perimeter of Rectangle 3: \( P_3 = 2(2b - a) + 2(2a - 3b) = 4b - 2a + 4a - 6b = 2a - 2b \)

4. **Set up the given conditions:**
   - Rectangle 1 has 20 cm more perimeter than Rectangle 2:
     \[
     2a = 2b + 20
     \]
   - Rectangle 2 has 16 cm more perimeter than Rectangle 3:
     \[
     2b = 2a - 2b + 16
     \]

5. **Solve the system of equations:**
   - From \( 2a = 2b + 20 \):
     \[
     a = b + 10
     \]
   - Substitute \( a = b + 10 \) into \( 2b = 2a - 2b + 16 \):
     \[
     2b = 2(b + 10) - 2b + 16
     \]
     \[
     2b = 2b + 20 - 2b + 16
     \]
     \[
     2b = 36
     \]
     \[
     b = 18
     \]
   - Substitute \( b = 18 \) back into \( a = b + 10 \):
     \[
     a = 18 + 10 = 28
     \]

6. **Calculate the area of the original sheet:**
   \[
   \text{Area} = a \times b = 28 \times 18 = 504 \, \text{cm}^2
   \]

The final answer is \( \boxed{504 \, \text{cm}^2} \).