In the 4 × 4 array shown below, each cell of the first three rows has either a…

Key idea: each cell's number equals the count of its immediate neighbors that are not a cross (X). Using the given X positions and numbers in the first three rows, we can determine each bottom cell directly.

• Bottom-left cell (row 4, column 1): its neighbors are the cell above (3,1)=3 (not X), the diagonal above-right (3,2)=X, and the adjacent bottom cell (4,2) which is not X. So it has exactly 2 non-X neighbors and its value is 2.

• Cell (row 4, column 2): neighbors are (3,1)=3, (3,2)=X, (3,3)=6, (4,1)=2 (just found), and (4,3) (unknown but not X). Only one neighbor is X, so out of 5 neighbors there are 4 non-X neighbors; its value is 4.

• Cell (row 4, column 3): neighbors are (3,2)=X, (3,3)=6, (3,4)=X, (4,2)=4, and (4,4) (unknown but not X). Two neighbors are X, so out of 5 neighbors there are 3 non-X neighbors; its value is 3.

• Bottom-right cell (row 4, column 4): neighbors are (3,3)=6, (3,4)=X, and (4,3)=3. Exactly one neighbor is X, so out of 3 neighbors there are 2 non-X neighbors; its value is 2.

Sum of the four bottom numbers: 2 + 4 + 3 + 2 = 11. Therefore the required sum is 11.