A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?

[i]Proposed by Evan Chen[/i]
1. **Determine the side length of the equilateral triangle:**
   - Given the area of the equilateral triangle is \(16\sqrt{3}\).
   - The formula for the area of an equilateral triangle with side length \(s\) is:
     \[
     \text{Area} = \frac{s^2 \sqrt{3}}{4}
     \]
   - Set the given area equal to the formula and solve for \(s\):
     \[
     \frac{s^2 \sqrt{3}}{4} = 16\sqrt{3}
     \]
   - Divide both sides by \(\sqrt{3}\):
     \[
     \frac{s^2}{4} = 16
     \]
   - Multiply both sides by 4:
     \[
     s^2 = 64
     \]
   - Take the square root of both sides:
     \[
     s = 8
     \]

2. **Calculate the perimeter of the equilateral triangle:**
   - The perimeter \(P\) of an equilateral triangle is given by:
     \[
     P = 3s
     \]
   - Substitute \(s = 8\):
     \[
     P = 3 \times 8 = 24
     \]

3. **Determine the side length of the square:**
   - The square has the same perimeter as the equilateral triangle, which is 24.
   - Let the side length of the square be \(a\).
   - The perimeter of the square is given by:
     \[
     4a = 24
     \]
   - Solve for \(a\):
     \[
     a = \frac{24}{4} = 6
     \]

4. **Calculate the area of the square:**
   - The area \(A\) of a square with side length \(a\) is:
     \[
     A = a^2
     \]
   - Substitute \(a = 6\):
     \[
     A = 6^2 = 36
     \]

The final answer is \(\boxed{36}\).