Difference between revisions of "2004 AMC 10A Problems/Problem 16"
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<math> \mathrm{(A) \ } 12 \qquad \mathrm{(B) \ } 15 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{(D) \ } 19\qquad \mathrm{(E) \ } 20 </math> | <math> \mathrm{(A) \ } 12 \qquad \mathrm{(B) \ } 15 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{(D) \ } 19\qquad \mathrm{(E) \ } 20 </math> | ||
| − | ==Solution== | + | ==Solution 1== |
There are: | There are: | ||
*<math>1</math> of the <math>1\times 1</math> squares containing the black square, | *<math>1</math> of the <math>1\times 1</math> squares containing the black square, | ||
| Line 15: | Line 15: | ||
Thus, the answer is <math>1+4+9+4+1=19\Rightarrow \boxed{\mathrm{(D)}}</math>. | Thus, the answer is <math>1+4+9+4+1=19\Rightarrow \boxed{\mathrm{(D)}}</math>. | ||
| + | |||
| + | ==Solution 2== | ||
| + | We use complementary counting. | ||
| + | There are only 2x2 and 1x1 squares that do not contain the black square. | ||
| + | Counting, there are <math>12</math> 2x2, and <math>25-1 = 24</math> 1x1 squares that do not contain the black square. | ||
| + | That gives <math>12+24=36</math> squares that don't contain it. | ||
| + | There are a total of <math>25+16+9+4+2+1 = 55</math> squares possible, therefore there are <math>55-36 = 19</math> squares that do not contain the black square, which is <math>\boxed{\mathrm{(D)}}</math>. . | ||
==See also== | ==See also== | ||
Revision as of 16:52, 30 December 2010
Contents
Problem
The 
grid shown contains a collection of squares with sizes from 
to . How many of these squares contain the black center square?
Solution 1
There are:
of the 
squares containing the black square,
of the 
squares containing the black square,
of the 
squares containing the black square,- of the
squares containing the black square, - of the
squares containing the black square.
of the 
of the 
squares containing the black square,
of the 
squares containing the black square,
squares containing the black square,Thus, the answer is 
.
Solution 2
We use complementary counting.
There are only 2x2 and 1x1 squares that do not contain the black square.
Counting, there are 
2x2, and 
1x1 squares that do not contain the black square.
That gives 
squares that don't contain it.
There are a total of 
squares possible, therefore there are 
squares that do not contain the black square, which is 
. .
See also
| 2004 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 | ||
