Difference between revisions of "2013 AIME I Problems/Problem 4"

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== Problem 4 ==
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== Problem ==
 
In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is <math>\frac{1}{n}</math> , where ''n'' is a positive integer. Find ''n''.
 
In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is <math>\frac{1}{n}</math> , where ''n'' is a positive integer. Find ''n''.
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<asy>
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draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0));
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draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0));
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draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2));
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draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1));
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draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1));
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size(100);</asy>
  
 
== Solution ==
 
== Solution ==
When the array appears the same after a 90 degree rotation, the top formation must look the same as the right formation, which looks the same as the bottom one, which looks the same as the right one. There are four of the same configuration. There are not enough red squares for these to be all red, nor are there enough blue squares for there to be more than one blue square in each three-square formation. Thus there are 2 reds and 1 blue in each, and a blue in the center. There are 3 ways to choose which of the squares in the formation will be blue, leaving the other two red.
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When the array appears the same after a 90-degree rotation, the top formation must look the same as the right formation, which looks the same as the bottom one, which looks the same as the right one. There are four of the same configuration. There are not enough red squares for these to be all red, nor are there enough blue squares for there to be more than one blue square in each three-square formation. Thus there are 2 reds and 1 blue in each, and a blue in the center. There are 3 ways to choose which of the squares in the formation will be blue, leaving the other two red.
  
 
There are <math>\binom{13}{5}</math> ways to have 5 blue squares in an array of 13.  
 
There are <math>\binom{13}{5}</math> ways to have 5 blue squares in an array of 13.  
  
<math>\frac{3}{\binom{13}{5}}</math> = <math>\frac{1}{429}</math> , so ''n'' = 429.
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<math>\frac{3}{\binom{13}{5}}</math> = <math>\frac{1}{429}</math> , so ''<math>n</math>'' = <math>\boxed{429}</math>
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==Video Solution==
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https://www.youtube.com/watch?v=9way8JrtD04&t=555s
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~Shreyas S
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== See also ==
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{{AIME box|year=2013|n=I|num-b=3|num-a=5}}
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{{MAA Notice}}

Latest revision as of 20:04, 24 January 2021

Problem

In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\frac{1}{n}$ , where n is a positive integer. Find n.

[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100);[/asy]

Solution

When the array appears the same after a 90-degree rotation, the top formation must look the same as the right formation, which looks the same as the bottom one, which looks the same as the right one. There are four of the same configuration. There are not enough red squares for these to be all red, nor are there enough blue squares for there to be more than one blue square in each three-square formation. Thus there are 2 reds and 1 blue in each, and a blue in the center. There are 3 ways to choose which of the squares in the formation will be blue, leaving the other two red.

There are $\binom{13}{5}$ ways to have 5 blue squares in an array of 13.

$\frac{3}{\binom{13}{5}}$ = $\frac{1}{429}$ , so $n$ = $\boxed{429}$

Video Solution

https://www.youtube.com/watch?v=9way8JrtD04&t=555s ~Shreyas S

See also

2013 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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