Problem: In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find $x$. [](https://artofproblemsolving.com/wiki/index.php/File:AIME_1996_Problem_01.png) Solution: We have to be clever with our system. Here, we know $x+19+96 = S$ where $S$ is the sum rule. We also know $x+y+1=S$, meaning $x = S-(y+1) = S-(115)$. We conclude that $y = 114$. From the diagonal, we also have that $96+114+z=S$ where $z$ is the middle entry, meaning $210+z=S$ We finally have $x+z+w=S$ where $w$ is the box in the bottom right corner. What is $w$? Well, we know that $114+w+(S-19-z)=S$ meaning $w = z-95$ meaning $x+z+w = S \iff +2z-95 = S$. From here, we get that $z = \frac{S-x+95}{2}$, so $210+z=S \iff 420+S-x+95=2S \iff 515-x=S$. Solving our system, we find $2x = 515-115 \implies 2x = 400 \implies x = 200$.