2001 AMC 10 Problem 22

1. Read the square from the picture:
$$\begin{array}{ccc} v & 24 & w \\ 18 & x & y \\ 25 & z & 21 \end{array}$$
Every row, column and diagonal has the same total.
2. The left column has only one unknown, $v$: its total is $v + 18 + 25 = v + 43$. Compare it with other lines through $v$, so the $v$ cancels.
3. Top row against left column. Top row: $v + 24 + w$. Left column: $v + 43$. The totals are equal, so $24 + w = 43$ and $w = 19$.
4. Diagonal against left column. The diagonal from top left to bottom right is $v + x + 21$. Setting it equal to $v + 43$ gives $x + 21 = 43$, so $x = 22$.
5. Find the common total. The other diagonal is $w + x + 25 = 19 + 22 + 25 = 66$. So every line adds to $66$.
6. Bottom row: $25 + z + 21 = 66$, so $z = 20$.
7. Middle row: $18 + 22 + y = 66$, so $y = 26$.
8. Check with a column: the right column is $w + y + 21 = 19 + 26 + 21 = 66$. It works.
9. So $y + z = 26 + 20 = 46$.

The answer is $\boxed{\textbf{(D)}\ 46}$.

A magic square gives many more equations than you need. The quick way is to set two lines equal when they share an unknown: that unknown appears on both sides and cancels, leaving a number. Two such comparisons plus one line to find the common total are enough here.

In every $3 \times 3$ magic square, the center number is one third of the common total. Once step 4 gives $x = 22$, the total is $3 \times 22 = 66$. Then the middle row gives $y = 66 - 18 - 22 = 26$, and the middle column gives $z = 66 - 24 - 22 = 20$, so $y + z = 46$.