A $$4\times4$$ anti-magic square is an arrangement of the numbers $$1$$ to $$16$$ in a square so that the totals of each of the four rows, four columns and the two diagonals are ten consecutive numbers in some order. The diagram shows an incomplete anti-magic square. When it is completed, the number in the position of $$*$$ is

The known fourth column has sum $$30$$ and the known fourth row has sum $$31$$, so the ten line sums must be the consecutive values $$30$$ through $$39$$. Testing the remaining unused values in the diagonal and row constraints forces the entry at $$*$$ to be $$16$$. The completed construction then satisfies the required ten consecutive line sums.