2025 AIME I Problems/Problem 10
Problem
The 
cells of a 
grid are filled in using the numbers 
through 
so that each row contains different numbers, and each of the three 
blocks heavily outlined in the example below contains 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]](http://latex.artofproblemsolving.com/d/f/3/df3daade257e4001f968e9c379a01ae9c94e8883.png)
The number of different ways to fill such a grid can be written as 
where 
, 
, 
, and 
are distinct prime numbers and 
, 
, 
, 
are positive integers. Find 
.
See also
| 2025 AIME I (Problems • Answer Key • Resources) | ||
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Followed by Problem 11 | |
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