The <img alt="$27$" height="14" src="https://latex.artofproblemsolving.com/1/0/5/1051e7d41ec0b8bd688a872be17538413f4ee97b.png" width="20" /> cells of a <img alt="$3 \times 9$" height="15" src="https://latex.artofproblemsolving.com/d/c/d/dcd353e0943439deb6742698401d1c2a6ba4cd5c.png" width="42" /> grid are filled in using the numbers <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" /> through <img alt="$9$" height="12" src="https://latex.artofproblemsolving.com/b/f/2/bf2c9074b396e3af0dea52d792660eea1c77f10f.png" width="8" /> so that each row contains <img alt="$9$" height="12" src="https://latex.artofproblemsolving.com/b/f/2/bf2c9074b396e3af0dea52d792660eea1c77f10f.png" width="8" /> different numbers, and each of the three <img alt="$3 \times 3$" height="15" src="https://latex.artofproblemsolving.com/d/2/7/d278803d72fcc493afb1559174a483aa7f41d143.png" width="42" /> blocks heavily outlined in the example below contains <img alt="$9$" height="12" src="https://latex.artofproblemsolving.com/b/f/2/bf2c9074b396e3af0dea52d792660eea1c77f10f.png" width="8" /> different numbers, as in the first three rows of a Sudoku puzzle. <img alt="[asy] unitsize(20); add(grid(9,3)); draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); real a = 0.5; label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a)); label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a)); label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a)); [/asy]" height="105" src="https://latex.artofproblemsolving.com/d/f/3/df3daade257e4001f968e9c379a01ae9c94e8883.png" width="305" /> The number of different ways to fill such a grid can be written as <img alt="$p^a \cdot q^b \cdot r^c \cdot s^d$" height="21" src="https://latex.artofproblemsolving.com/f/e/6/fe6124c9608bf33d129b8ae0ffc2b2e7fd8f62e5.png" width="106" /> where <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" />, <img alt="$q$" height="11" src="https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png" width="8" />, <img alt="$r$" height="8" src="https://latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png" width="8" />, and <img alt="$s$" height="8" src="https://latex.artofproblemsolving.com/f/3/7/f37bba504894945c07a32f5496d74299a37aa51c.png" width="8" /> are distinct prime numbers and <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" />, <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" />, <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" />, <img alt="$d$" height="12" src="https://latex.artofproblemsolving.com/9/6/a/96ab646de7704969b91c76a214126b45f2b07b25.png" width="9" /> are positive integers. Find <img alt="$p \cdot a + q \cdot b + r \cdot c + s \cdot d$" height="18" src="https://latex.artofproblemsolving.com/2/4/c/24c7f5ec2137682b717e7c25c48cf3a76fe90dca.png" width="191" />.

Question No: 407

The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle.

$[asy] unitsize(20); add(grid(9,3)); draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); real a = 0.5; label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a)); label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a)); label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a)); [/asy]$

The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$ , $q$ , $r$ , and $s$ are distinct prime numbers and $a$ , $b$ , $c$ , $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$ .

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Tags: Combinatorics

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