As the Kubiks head out of town for vacation, Jerry takes the first driving shift while Hannah and most of the kids settle down to read books they brought along.  Tony does not feel like reading, so Alexis gives him one of her math notebooks and Tony gets to work solving some of the problems, and struggling over others.  After a while, Tony comes to a problem he likes from an old AMC 10 exam:
\begin{align*}&\text{Four distinct circles are drawn in a plane.  What is the maximum}\\&\quad\,\,\text{number of points where at least two of the circles intersect?}\end{align*}
Tony realizes that he can draw the four circles such that each pair of circles intersects in two points.  After careful doodling, Tony finds the correct answer, and is proud that he can solve a problem from late on an AMC 10 exam.

"Mom, why didn't we all get Tony's brain?" Wendy inquires before turning he head back into her favorite Harry Potter volume (the fifth year).

Joshua leans over to Tony's seat to see his brother's work.  Joshua knows that Tony has not yet discovered all the underlying principles behind the problem, so Joshua challenges, "What if there are a dozen circles?"

Tony gets to work on Joshua's problem of finding the maximum number of points of intersections where at least two of the twelve circles in a plane intersect.  What is the answer to this problem?
To solve the problem of finding the maximum number of points where at least two of the twelve circles intersect, we need to consider the number of intersection points formed by pairs of circles.

1. **Understanding Circle Intersections**:
   - When two distinct circles intersect, they can intersect at most in 2 points.
   - Therefore, for each pair of circles, there are at most 2 intersection points.

2. **Counting Pairs of Circles**:
   - We need to determine the number of ways to choose 2 circles out of 12. This is a combination problem.
   - The number of ways to choose 2 circles from 12 is given by the binomial coefficient:
     \[
     \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66
     \]

3. **Calculating Total Intersection Points**:
   - Since each pair of circles can intersect at most in 2 points, the total number of intersection points is:
     \[
     2 \times \binom{12}{2} = 2 \times 66 = 132
     \]

Thus, the maximum number of points where at least two of the twelve circles intersect is \( \boxed{132} \).