6. (10 points) As shown in the figure, three circles intersect to form seven regions. Place the integers $1 \sim 7$ into these seven regions, such that the sum of the four numbers within each circle is equal. What is the maximum value of this sum?
19 .
【Analysis】Since it is required that the sum of the four numbers in each circle is equal, and when the maximum sum is achieved, 7 is the largest, so we place 7 in the center. The remaining 3 larger numbers are $6, 5, 4$, which are placed in the overlapping parts of the two circles. The remaining numbers are $1, 2, 3$: 1 is combined with 7, 6, 5; 2 is combined with $7, 6, 4$; 3 is combined with $7, 5, 4$. Thus, the sum of the four numbers in each circle is 19. According to this, the solution can be derived.

【Solution】According to the analysis, the maximum sum is 19.

Therefore, the answer is: 19.