Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 15"
| Line 12: | Line 12: | ||
Therefore the total numbers of squares that can have exactly one red in each column and exactly one red in each row and one red in each row is exactly <math>4!</math> | Therefore the total numbers of squares that can have exactly one red in each column and exactly one red in each row and one red in each row is exactly <math>4!</math> | ||
| + | |||
| + | Now we can use this information for the 3D problem by looking at each of these squares as levels of the cube starting with the first level that has 4! configurations. | ||
| + | |||
| + | [[Image:AIME_2006_P15c2.png|500px]] | ||
| + | |||
==See Also== | ==See Also== | ||
Revision as of 20:40, 22 November 2023
Contents
Problem
A 
cube is composed of 
unit cubes. The faces of 
unit cubes are colored red. An arrangement of the cubes is "intriguing" if there is exactly 
red unit cube in every 
rectangular box composed of 
unit cubes. Determine the number of "intriguing" colorings.
Solution
In order to solve this we must first look at the 2D problem:
In order to have exactly one red in each column and exactly one red in each row, one can select any square red in the first column, for the second column we can only chose from 3 to paint red, the third column we can only chose 2 and the last one we can only chose 1.
Therefore the total numbers of squares that can have exactly one red in each column and exactly one red in each row and one red in each row is exactly 
Now we can use this information for the 3D problem by looking at each of these squares as levels of the cube starting with the first level that has 4! configurations.
See Also
| Mock AIME 2 2006-2007 (Problems, Source) | ||
| Preceded by Problem 14 |
Followed by Last Question | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
