1. **State the problem:** We have a 3x3 magic square where every row, column, and diagonal sums to the same total. We need to find the values of $A$ and $B$ using the numbers provided: $-1, 0, 3, 4$ (since others are crossed out). 2. **Set up the magic sum:** Let the magic sum be $S$. Each row, column, and diagonal must sum to $S$. 3. **Write equations for rows:** - Top row: $-3 + A + B = S$ - Middle row: $2 + \text{middle} + (-2) = S$; middle is unknown, but $2 + (-2) = 0$, so middle element must be $S$. - Bottom row: $1 + (-4) + \text{bottom right} = S$; $1 - 4 = -3$, so bottom right element must be $S + 3$. 4. **Write equations for columns:** - First column: $-3 + 2 + 1 = 0$; so $S=0$ if the magic sum is consistent. 5. **Check if $S=0$ works:** - Top row: $-3 + A + B = 0 \Rightarrow A + B = 3$ - Middle row: $2 + \text{middle} + (-2) = 0 \Rightarrow \text{middle} = 0$ - Bottom row: $1 + (-4) + \text{bottom right} = 0 \Rightarrow \text{bottom right} = 3$ 6. **Fill known values:** - Middle element is $0$ - Bottom right element is $3$ 7. **Check columns:** - Second column: $A + 0 + (-4) = 0 \Rightarrow A - 4 = 0 \Rightarrow A = 4$ - Third column: $B + (-2) + 3 = 0 \Rightarrow B + 1 = 0 \Rightarrow B = -1$ 8. **Verify diagonals:** - Main diagonal: $-3 + 0 + 3 = 0$ - Other diagonal: $B + 0 + 1 = -1 + 0 + 1 = 0$ 9. **Check if $A=4$ and $B=-1$ are in the allowed numbers:** - $4$ is allowed - $-1$ is allowed **Final answer:** $$A = 4, \quad B = -1$$