Grids & Bubbles

It seems like there isn't enough information to get started, since we're only given the absolute differences. There is likely some case checking involved, but we want to minimize the number of cases. Certainly, looking at $8!$ cases isn't ideal.

Which part of the puzzle contains the most information? If the absolute difference of 2 numbers is high, then this means that the numbers are very restrictive (must be very large and very small). Hence, let's focus our attention on the bottom middle (BM) and bottom left (BL) squares, which have a difference of 6. The possible pairs are: $1-7, 2-8, 3-9$ .

Let's label the squares by their row (Top / Center / Bottom) and their column (Left / Middle / Right).

Case 1: $BM = 1$ .
Then $BL = 7$ which contradicits the distinct digits condition.

Case 2: $BM = 2$ .
Then $BR = 7$ , which contradicts the distinct digits condition.

Case 3: $BM = 3$ .
Then $BL = 9, BR = 8$ .
Then $CL = 5, CR = 6$ .
But then TR would be 3 or 9, which contradicts the distinct digits condition.

Case 4: $BM = 7$ .
This contradicts the distinct digits condition.

Case 5: $BM = 8$ .
Then $BL = 2, BR = 3$ .
Then $CL = 6$ .
Then $TL = 5$ (since it is not 7).
The digits we have left are 1, 4, 9. Since $|TR - TM| = 5, |TR - CR| = 3$ , hence we must have $TR = 4, TM = 9, CR = 3$ .

Case 6: $BM = 9$
Then $BL = 3 , BR = 4$ .
Then $CL = 7$ , which contradicts the distinct digits condition.

In conclusion, only case 5 is possible, and we get $\overline{ABCD} = {2161}$ .

just check 8 possible cases for placing 9 and now as 9 is largest no. we know the two numbers in its neigbourhood and thus every case proceeds smoothly