6. (10 points) $n$ pieces of paper, each written with 3 different positive integers not greater than $n$, any 2 pieces of paper have exactly one number in common, find the sum of all numbers written on the pieces of paper.


【Analysis】From the problem, each number appears 3 times, $n \geqslant 2 k+1=7$, further analysis shows that $n=7$. At this point, the numbers on the 7 cards are $1-7$, each appearing 3 times, and the sum of the numbers is 84.

【Solution】Let $a$ be the number that appears the most. There are a total of $k$ cards, so these $k$ cards have $2 k+1$ different numbers. Since each number is no greater than $n$, we have $2 k+1 \leqslant n$. Therefore, $k7$, the three cards with 1 have 7 different numbers. Since $n>7$, there is another number that does not appear on these three cards, denoted as $b$.

The card with $b$ has 3 numbers. This card shares exactly one number with each of the three cards with $a$, and this number is neither 1 nor $b$. However, the card with $b$ has only 2 other numbers besides $b$, making it impossible to share a number with each of the three cards with 1. Therefore, $n=7$. Each number appears exactly on three cards, and the sum of all the numbers written is $3 \times(1+2+\cdots+7)=3 \times 28=84$.

Here is an example: $1,2,3 ; 1,4,5 ; 1,6,7 ; 2,4,6 ; 2,5,7 ; 3,4,7 ; 3,5,6$.