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Difference between revisions of "2025 AIME I Problems/Problem 10"

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==Problem==
  
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The <math>27</math> cells of a <math>3 \times 9</math> grid are filled in using the numbers <math>1</math> through <math>9</math> so that each row contains <math>9</math> different numbers, and each of the three <math>3 \times 3</math> blocks heavily outlined in the example below contains <math>9</math> different numbers, as in the first three rows of a Sudoku puzzle.
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<asy>
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unitsize(20);
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add(grid(9,3));
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draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2));
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draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2));
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real a = 0.5;
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label("5",(a,a));
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label("6",(1+a,a));
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label("1",(2+a,a));
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label("8",(3+a,a));
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label("4",(4+a,a));
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label("7",(5+a,a));
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label("9",(6+a,a));
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label("2",(7+a,a));
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label("3",(8+a,a));
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label("3",(a,1+a));
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label("7",(1+a,1+a));
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label("9",(2+a,1+a));
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label("5",(3+a,1+a));
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label("2",(4+a,1+a));
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label("1",(5+a,1+a));
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label("6",(6+a,1+a));
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label("8",(7+a,1+a));
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label("4",(8+a,1+a));
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label("4",(a,2+a));
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label("2",(1+a,2+a));
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label("8",(2+a,2+a));
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label("9",(3+a,2+a));
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label("6",(4+a,2+a));
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label("3",(5+a,2+a));
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label("1",(6+a,2+a));
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label("7",(7+a,2+a));
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label("5",(8+a,2+a));
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</asy>
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The number of different ways to fill such a grid can be written as <math>p^a \cdot q^b \cdot r^c \cdot s^d</math> where <math>p</math>, <math>q</math>, <math>r</math>, and <math>s</math> are distinct prime numbers and <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are positive integers. Find <math>p \cdot a + q \cdot b + r \cdot c + s \cdot d</math>.
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==See also==
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{{AIME box|year=2025|num-b=9|num-a=11|n=I}}
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{{MAA Notice}}

Revision as of 19:50, 13 February 2025

Problem

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 $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.

See also

2025 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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