An equilateral triangle of side $x$ has its vertices on the sides of a square side $1$. What are the possible values of $x$?	
1. **Define the problem geometrically:**
   We are given a square \(ABCD\) with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\). We need to find the possible side lengths \(x\) of an equilateral triangle whose vertices lie on the sides of this square.

2. **Positioning the vertices of the equilateral triangle:**
   Let's denote the vertices of the equilateral triangle as \(P\), \(Q\), and \(R\). We need to consider different cases based on the positioning of these vertices on the sides of the square.

3. **Case 1: One vertex on each side:**
   Suppose one vertex \(P\) is on side \(AB\), another vertex \(Q\) is on side \(BC\), and the third vertex \(R\) is on side \(CD\). We need to find the coordinates of these points and the side length \(x\).

4. **Using coordinate geometry:**
   Let \(P(a,0)\) be on \(AB\), \(Q(1,b)\) be on \(BC\), and \(R(c,1)\) be on \(CD\). Since \(PQR\) is an equilateral triangle, the distances \(PQ\), \(QR\), and \(RP\) must all be equal to \(x\).

5. **Distance formula application:**
   Using the distance formula, we have:
   \[
   PQ = \sqrt{(1-a)^2 + b^2} = x
   \]
   \[
   QR = \sqrt{(1-c)^2 + (1-b)^2} = x
   \]
   \[
   RP = \sqrt{c^2 + (1-0)^2} = x
   \]

6. **Simplifying the equations:**
   We need to solve these equations simultaneously to find the possible values of \(x\). However, this system of equations is complex and may not yield a simple analytical solution. Instead, we can use geometric properties and symmetry to simplify our approach.

7. **Geometric properties:**
   Consider the symmetry of the problem. The maximum side length of the equilateral triangle will occur when the vertices are positioned such that the triangle is inscribed in the square. The side length of the inscribed equilateral triangle can be found using the relationship between the side length of the square and the side length of the inscribed equilateral triangle.

8. **Inscribed equilateral triangle:**
   The side length \(x\) of an equilateral triangle inscribed in a square of side length 1 can be derived using trigonometry. The height \(h\) of the equilateral triangle is given by:
   \[
   h = \frac{\sqrt{3}}{2} x
   \]
   Since the height of the equilateral triangle must fit within the side length of the square, we have:
   \[
   \frac{\sqrt{3}}{2} x = 1 \implies x = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
   \]

9. **Conclusion:**
   The possible value of \(x\) for an equilateral triangle inscribed in a square of side length 1 is \(\frac{2\sqrt{3}}{3}\).

The final answer is \(\boxed{\frac{2\sqrt{3}}{3}}\).