Given is the set of nine numbers that relate to each other in a certain way.…

Concept: In this type of number-grid reasoning puzzle, the entries in each row are not analysed individually; instead, the SUM of the entries in a row produces a value, and the sequence of these row-sums across the grid(s) follows a specific numerical rule (very often a run of consecutive prime numbers). Once that governing rule is confirmed on the fully-known grid, it is applied to find the single missing entry in the incomplete grid.

Working:

1. Add the entries in each row of the first (fully known) grid: 4 + 3 + 4 = 11, 5 + 1 + 7 = 13, 4 + 4 + 9 = 17.

2. 11, 13 and 17 are three consecutive prime numbers — each of them is prime, and no prime number lies between successive terms.

3. Add the entries in the first two rows of the second grid: 12 + 21 + 28 = 61 and 21 + 12 + 34 = 67. Both are prime, continuing the same consecutive-prime progression seen in the first grid.

4. Since the row-sums must keep advancing through consecutive prime numbers, the third row's sum must be the next prime number after 67, which is 71.

5. Set up the equation for the third row of the second grid: 17 + 14 + X = 71.

6. Solve for X: X = 71 − (17 + 14) = 71 − 31 = 40.

Cross-check: 71 is prime (it has no factor among 2, 3, 5 or 7, which covers every prime up to its square root), and none of 68, 69 or 70 is prime, so 71 really is the very next prime after 67. Re-adding 17 + 14 + 40 = 71 also confirms the result.