Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?
1. **Understanding the Problem:**
   We need to determine for which positive integers \( n \) it is possible to place \( n \) distinct circles with the same diameter on the plane such that each circle touches at least three other circles.

2. **Basic Configuration:**
   Let's start by considering small configurations. A single circle cannot touch any other circle. Two circles can touch each other, but neither touches three circles. Three circles can be arranged in a triangle, but each touches only two others. Four circles can form a square, but again, each touches only two others.

3. **Hexagonal Arrangement:**
   The smallest configuration where each circle touches at least three others is a hexagon with a central circle. This configuration consists of 7 circles: one in the center and six surrounding it. Each of the six surrounding circles touches the central circle and two adjacent circles, making a total of three touches per circle.

4. **Larger Configurations:**
   We can extend this idea by adding more circles in a hexagonal pattern. For example, a "7-hexagon" consists of 7 circles, and a "10-hexagon" consists of 10 circles. These configurations can be repeated and combined to form larger structures.

5. **Generalizing the Pattern:**
   To generalize, we can use the idea of tiling the plane with hexagons. Each hexagon consists of 7 circles, and we can add more hexagons to increase the number of circles while maintaining the property that each circle touches at least three others.

6. **Constructing 2012 Circles:**
   To place 2012 circles, we can use 286 pieces of 7-hexagons and one 10-hexagon:
   \[
   286 \times 7 + 10 = 2002 + 10 = 2012
   \]
   This ensures that each circle touches at least three others.

7. **General Case:**
   For a general positive integer \( n \), we need to determine if \( n \) can be expressed as a combination of 7-hexagons and possibly a few additional circles to ensure each circle touches at least three others. The key is to find a way to tile the plane such that each circle in the configuration touches at least three others.

8. **Conclusion:**
   Based on the hexagonal tiling method, it is possible to place \( n \) distinct circles with the same diameter on the plane such that each circle touches at least three others for sufficiently large \( n \). Specifically, \( n \) must be a combination of multiples of 7 plus a few additional circles to complete the configuration.

The final answer is \( \boxed{ n } \) can be any positive integer that can be expressed as a combination of multiples of 7 plus a few additional circles to complete the configuration.