Glad to know the previous puzzle, which was the first puzzle I posted in Puzzling, was warmly welcomed (Thank you!), and an optimal solution was found. Inspired by the comments there, here is the Version $2$ of the puzzle.
In fact, most of the things are unchanged. We still have this $3 \times 3$ grid, which $9$ distinct prime numbers $P_1, P_2, ..., P_9$ are to be filled in. And there are $8$ sums: $3$ horizontal, $3$ vertical and $2$ diagonal, and they are named $S_1, S_2, ..., S_8$. All the requirements in the first version still hold here, which mean:
- $P_1, P_2, ..., P_9, S_1, S_2, ..., S_8$ are all distinct prime numbers (i.e. there are totally $17$ different prime numbers).
But this time, one more additional requirement:
- The grand total $P_1 + P_2 + ... + P_9 + S_1 + S_2 + ... + S_8$ also has to be a prime number.
The challenge: To minimize the grand total.
With the additional requirement, some solutions satisfying the previous puzzle do not satisfy this version. And, the optimal solution will be different.
Below is one possible solution I come up with, which has a grand total of $601$, but it is not the optimal solution:
Feel free to have a try!