2001 AMC 10 Problems/Problem 22
Contents
Problem
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by 
, 
, 
, 
, and 
. Find 
.
![[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$w$",(2.5,2.5));[/asy]](http://latex.artofproblemsolving.com/4/e/2/4e24d4a82884d1873aa08b1502e6996d8292e7ed.png)
Solutions
Solution 1
We know that 
, so we could find one variable rather than two.
![[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]](http://latex.artofproblemsolving.com/a/c/9/ac93cbfe3fce005ca5ee9103f9f203476e189db6.png)
![[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$20$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]](http://latex.artofproblemsolving.com/1/2/6/1261087702926461e26052453481b55bb6c07a10.png)
The sum per row is 
.
Thus 
.
Since we needed 
and we know 
, 
.
Solution 2
The magic sum is determined by the bottom row. 
.
Solving for :
.
To find our answer, we need to find 
. 
.
