# 2006 iTest Problems/Problem 19

## Problem

Find the sum of $w + x + y + z$ by solving the Sudoku puzzle below.

1 _ _ | 3 5 8 | _ _ 6
4 _ _ | _ _ _ | _ x 8
_ _ 9 | _ 1 _ | 7 _ _
---------------------
_ z _ | 1 _ _ | _ 5 _
_ _ 3 | 2 _ 4 | 8 _ _
_ 2 _ | w _ 9 | _ _ _
---------------------
_ _ 6 | _ 2 _ | 9 _ _
3 _ _ | _ y _ | _ _ 1
2 _ _ | 8 4 3 | _ _ 7


$\textbf{(A) }7\qquad \textbf{(B) }8\qquad \textbf{(C) }9\qquad \textbf{(D) }10\qquad \textbf{(E) }11\qquad \textbf{(F) }12\qquad \textbf{(G) }13\qquad \textbf{(H) }14\qquad \textbf{(I) }15\qquad \textbf{(J) }16\qquad \textbf{(K) }17\qquad \textbf{(L) }18\qquad$ $\textbf{(M) }19\qquad \textbf{(N) }20\qquad \textbf{(O) }21\qquad \textbf{(P) }22\qquad \textbf{(Q) }23\qquad \textbf{(R) }24\qquad \textbf{(S) }25$

## Solution

$[asy] unitsize(2.5); draw((0,0)--(0,90)--(90,90)--(90,0)--cycle,linewidth(2)); draw((10,0)--(10,90)); draw((20,0)--(20,90)); draw((30,0)--(30,90),linewidth(2)); draw((40,0)--(40,90)); draw((50,0)--(50,90)); draw((60,0)--(60,90),linewidth(2)); draw((70,0)--(70,90)); draw((80,0)--(80,90)); draw((0,10)--(90,10)); draw((0,20)--(90,20)); draw((0,30)--(90,30),linewidth(2)); draw((0,40)--(90,40)); draw((0,50)--(90,50)); draw((0,60)--(90,60),linewidth(2)); draw((0,70)--(90,70)); draw((0,80)--(90,80)); label("1",(5,85)); label("7",(15,85)); label("2",(25,85)); label("3",(35,85)); label("5",(45,85)); label("8",(55,85)); label("4",(65,85)); label("9",(75,85)); label("6",(85,85)); label("4",(5,75)); label("3",(15,75)); label("5",(25,75)); label("9",(35,75)); label("7",(45,75)); label("6",(55,75)); label("1",(65,75)); label("2",(75,75)); label("8",(85,75)); label("6",(5,65)); label("8",(15,65)); label("9",(25,65)); label("4",(35,65)); label("1",(45,65)); label("2",(55,65)); label("7",(65,65)); label("3",(75,65)); label("5",(85,65)); label("9",(5,55)); label("6",(15,55)); label("4",(25,55)); label("1",(35,55)); label("8",(45,55)); label("7",(55,55)); label("3",(65,55)); label("5",(75,55)); label("2",(85,55)); label("5",(5,45)); label("1",(15,45)); label("3",(25,45)); label("2",(35,45)); label("6",(45,45)); label("4",(55,45)); label("8",(65,45)); label("7",(75,45)); label("9",(85,45)); label("7",(5,35)); label("2",(15,35)); label("8",(25,35)); label("5",(35,35)); label("3",(45,35)); label("9",(55,35)); label("6",(65,35)); label("1",(75,35)); label("4",(85,35)); label("8",(5,25)); label("5",(15,25)); label("6",(25,25)); label("7",(35,25)); label("2",(45,25)); label("1",(55,25)); label("9",(65,25)); label("4",(75,25)); label("3",(85,25)); label("3",(5,15)); label("4",(15,15)); label("7",(25,15)); label("6",(35,15)); label("9",(45,15)); label("5",(55,15)); label("2",(65,15)); label("8",(75,15)); label("1",(85,15)); label("2",(5,5)); label("9",(15,5)); label("1",(25,5)); label("8",(35,5)); label("4",(45,5)); label("3",(55,5)); label("5",(65,5)); label("6",(75,5)); label("7",(85,5)); [/asy]$

Therefore, $$w+x+y+z = \boxed{\textbf{(P) }22}$$