Difference between revisions of "2012 AMC 10A Problems/Problem 20"
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| − | ==Problem | + | ==Problem== |
A <math>3</math> x <math>3</math> square is partitioned into <math>9</math> unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated <math>90\,^{\circ}</math> clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black? | A <math>3</math> x <math>3</math> square is partitioned into <math>9</math> unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated <math>90\,^{\circ}</math> clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black? | ||
<math> \textbf{(A)}\ \frac{49}{512}\qquad\textbf{(B)}\ \frac{7}{64}\qquad\textbf{(C)}\ \frac{121}{1024}\qquad\textbf{(D)}\ \frac{81}{512}\qquad\textbf{(E)}\ \frac{9}{32} </math> | <math> \textbf{(A)}\ \frac{49}{512}\qquad\textbf{(B)}\ \frac{7}{64}\qquad\textbf{(C)}\ \frac{121}{1024}\qquad\textbf{(D)}\ \frac{81}{512}\qquad\textbf{(E)}\ \frac{9}{32} </math> | ||
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| + | == Solution == | ||
| + | First, the middle square is always black because it is rotated onto itself. Then we can consider the corners and edges separately. Let's first just consider the number of ways we can color the corners. There is <math>1</math> case with all black squares. There are four cases with one white square and all <math>4</math> work. There are six cases with two white squares, but only the <math>2</math> with the white squares opposite from each other work. There are no cases with three white squares or four white squares. Then the total number of ways to color the corners is <math>1+4+2=7</math>. In essence, the edges work the same way, so there are also <math>7</math> ways to color them. The number of ways to fit the conditions over the number of ways to color the squares is | ||
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| + | <cmath>\frac{7\times7}{2^9}=\boxed{\textbf{(A)}\ \frac{49}{512}}</cmath> | ||
| + | |||
| + | == See Also == | ||
| + | |||
| + | {{AMC10 box|year=2012|ab=A|num-b=19|num-a=21}} | ||
Revision as of 00:27, 9 February 2012
Problem
A 
x square is partitioned into 
unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated 
clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
Solution
First, the middle square is always black because it is rotated onto itself. Then we can consider the corners and edges separately. Let's first just consider the number of ways we can color the corners. There is 
case with all black squares. There are four cases with one white square and all 
work. There are six cases with two white squares, but only the 
with the white squares opposite from each other work. There are no cases with three white squares or four white squares. Then the total number of ways to color the corners is 
. In essence, the edges work the same way, so there are also 
ways to color them. The number of ways to fit the conditions over the number of ways to color the squares is
![\[\frac{7\times7}{2^9}=\boxed{\textbf{(A)}\ \frac{49}{512}}\]](http://latex.artofproblemsolving.com/d/1/2/d12be61744c1adb4e336be847dcbf4e9a9ea7d06.png)
See Also
| 2012 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
