1. **Problem Statement:** Complete the 3x3 magic square such that the sum of the numbers in each row, column, and diagonal equals 21. Given partial magic square: | 10 | | 3 | | | 7 | | | | | | 2. **Formula and Rules:** - In a 3x3 magic square, the sum of each row, column, and diagonal is the magic constant, here 21. - Each row, column, and diagonal must sum to 21. 3. **Step-by-step solution:** - Let the missing cells be labeled as follows: | 10 | a | 3 | | b | 7 | c | | d | e | f | - From the first row sum: $$10 + a + 3 = 21 \implies a = 21 - 13 = 8$$ - From the second row sum: $$b + 7 + c = 21 \implies b + c = 14$$ - From the third row sum: $$d + e + f = 21$$ - From the first column sum: $$10 + b + d = 21 \implies b + d = 11$$ - From the second column sum: $$a + 7 + e = 21 \implies 8 + 7 + e = 21 \implies e = 6$$ - From the third column sum: $$3 + c + f = 21 \implies c + f = 18$$ - From the main diagonal (top-left to bottom-right): $$10 + 7 + f = 21 \implies f = 4$$ - From the other diagonal (top-right to bottom-left): $$3 + 7 + d = 21 \implies d = 11$$ - Now, from $b + d = 11$ and $d = 11$, we get: $$b + 11 = 11 \implies b = 0$$ - From $b + c = 14$ and $b = 0$, we get: $$0 + c = 14 \implies c = 14$$ - From $c + f = 18$ and $f = 4$, we get: $$14 + 4 = 18$$ which is consistent. - Finally, check the third row sum: $$d + e + f = 11 + 6 + 4 = 21$$ correct. 4. **Completed magic square:** | 10 | 8 | 3 | | 0 | 7 | 14 | | 11 | 6 | 4 | 5. **Verification:** - Each row, column, and diagonal sums to 21. **Final answer:** The completed magic square is: | 10 | 8 | 3 | | 0 | 7 | 14 | | 11 | 6 | 4 |