My initial question had lots of answers, as we can practically choose the middle square to be anything we want, and work outwards to the exterior edges, with the corners providing the flexibility needed.
So, as a bonus, I have added some more constraints. Same rules as before, with the outside sums only counting 2 sets of minuses.
8 7 3
--- --- --- ---
5 13 14 11 5
--- --- --- ---
7 11 10 13 7
--- --- --- ---
8 11 9 14 8
--- --- --- ---
5 6 9
Can you now solve the grid?
Answer
I think I have the answer
8 7 3
3 5 2 1
5 13 14 11 5
2 3 4 4
7 11 10 13 7
5 1 2 3
8 11 9 14 8
3 2 4 5
5 6 9
Reasoning
The 8 at the top left hand corner of the grid must be made by the sum of 3 and 5 or 4 and 4. If 5 is on the left, this forces the digit below to be 0 (which is not allowed). If 4 is on the left it forces the digit two below to be 6 (also not allowed). Hence, it must be 3 on the left and 5 on the right. From there, we can easily follow the grid around the edge and then deduce the inside part from remainders.
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