Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infinity, what is the ratio between the number of triangles and the number of squares?
1. **Understanding the Problem:**
   - We start with a regular hexagon.
   - From each edge of the hexagon, we construct a square.
   - We then form equilateral triangles by drawing edges between every pair of neighboring squares.
   - This process is continued symmetrically off to infinity.
   - We need to find the ratio between the number of triangles and the number of squares.

2. **Initial Configuration:**
   - A regular hexagon has 6 sides.
   - When we construct a square on each side of the hexagon, we add 6 squares.

3. **Forming Equilateral Triangles:**
   - Each pair of neighboring squares will form an equilateral triangle.
   - Since the hexagon has 6 sides, and each side has a neighboring square, we will form 6 equilateral triangles in this initial step.

4. **Next Layer:**
   - After the first step, we now have a 12-sided polygon (a dodecagon) formed by the outer edges of the squares.
   - We repeat the process: construct a square on each of the 12 sides of the dodecagon.
   - This adds 12 new squares.

5. **Forming Equilateral Triangles in the Next Layer:**
   - Each pair of neighboring squares on the dodecagon will form an equilateral triangle.
   - Since the dodecagon has 12 sides, we will form 12 equilateral triangles in this step.

6. **General Pattern:**
   - At each layer, the number of sides of the polygon is equal to the number of squares added in that layer.
   - The number of equilateral triangles formed is also equal to the number of sides of the polygon (or the number of squares added).

7. **Ratio Calculation:**
   - For each layer, the number of squares added is equal to the number of equilateral triangles formed.
   - Therefore, the ratio of the number of triangles to the number of squares is 1:1.

Conclusion:
The ratio between the number of triangles and the number of squares is $\boxed{1:1}$.