# 1996 AIME Problems/Problem 1

## 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 $x$.

## Solution

Let's make a table.

$$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table}\\\hline x&19&96\\\hline 1&a&b\\\hline c&d&e\\\hline \end{tabular}$$

$\begin{eqnarray*} x+19+96=x+1+c\Rightarrow c=19+96-1=114,\\ 114+96+a=x+1+114\Rightarrow a=x-95 \end{eqnarray*}$

$$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table in progress}\\\hline x&19&96\\\hline 1&x-95&b\\\hline 114&d&e\\\hline \end{tabular}$$

$\begin{eqnarray*}19+x-95+d=x+d-76=115+x\Rightarrow d=191,\\ 114+191+e=x+115\Rightarrow e=x-190 \end{eqnarray*}$ $$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table in progress}\\\hline x&19&96\\\hline 1&x-95&b\\\hline 114&191&x-190\\\hline \end{tabular}$$

$$3x-285=x+115\Rightarrow 2x=400\Rightarrow x=\boxed{200}$$

 1996 AIME (Problems • Answer Key • Resources) Preceded byFirst Problem Followed byProblem 2 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions