6. Each point on a circle is colored one of three colors: red, yellow, or blue, and all three colors are used. Now, take any $n$ points from the circle. If among these points, there always exist three points that form a triangle with all vertices of the same color and the triangle is obtuse, then the minimum possible value of $n$ is $\qquad$
6. 13 .

First, we state that \( n \geqslant 13 \). If \( n \geqslant 13 \), according to the pigeonhole principle, among these \( n \) points, there must be \(\left[\frac{13-1}{3}\right]+1=5\) points of the same color (let's assume they are red). Taking any 5 red points, it is easy to see that there always exists a diameter whose endpoints do not belong to these 5 points. According to the pigeonhole principle, among these 5 red points, there are always three points located on the same side of this diameter, and the triangle formed by these three points must be an obtuse triangle.

When \( n=12 \), consider the following scenario: select three inscribed squares \( A_{1} A_{2} A_{3} A_{4}, B_{1} B_{2} B_{3} B_{4} \), \( C_{1} C_{2} C_{3} C_{4} \), with the vertices of these three squares being red, yellow, and blue, respectively. In this case, all triangles with three vertices of the same color are right triangles, so \( n=12 \) does not meet the requirement.
In summary, the smallest possible value of \( n \) is 13.