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 1== | + | ==Solution== |
+ | ===Solution 1=== | ||
Since there are five types of squares: <math>1 \times 1, 2 \times 2, 3 \times 3, 4 \times 4,</math> and <math>5 \times 5.</math> We must find how many of each square contain the black shaded square in the center. | Since there are five types of squares: <math>1 \times 1, 2 \times 2, 3 \times 3, 4 \times 4,</math> and <math>5 \times 5.</math> We must find how many of each square contain the black shaded square in the center. | ||
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Thus, the answer is <math>1+4+9+4+1=19\Rightarrow\boxed{\mathrm{(D)}\ 19}</math>. | Thus, the answer is <math>1+4+9+4+1=19\Rightarrow\boxed{\mathrm{(D)}\ 19}</math>. | ||
− | ==Solution 2== | + | ===Solution 2=== |
We use complementary counting. There are only <math>2\times2</math> and <math>1\times1</math> squares that do not contain the black square. Counting, there are <math>12</math>-<math>2\times2</math> squares, and <math>25-1 = 24</math> <math>1\times1</math> 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+1 = 55</math> squares possible <math>(25</math> - <math>1\times1</math> squares <math>16</math> - <math>2\times2</math> squares <math>9</math> - <math>3\times3</math> squares <math>4</math> - <math>4\times4</math> squares and <math>1</math> - <math>5\times5</math> square), therefore there are <math>55-36 = 19</math> squares that contain the black square, which is <math>\boxed{\mathrm{(D)}\ 19}</math>. | We use complementary counting. There are only <math>2\times2</math> and <math>1\times1</math> squares that do not contain the black square. Counting, there are <math>12</math>-<math>2\times2</math> squares, and <math>25-1 = 24</math> <math>1\times1</math> 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+1 = 55</math> squares possible <math>(25</math> - <math>1\times1</math> squares <math>16</math> - <math>2\times2</math> squares <math>9</math> - <math>3\times3</math> squares <math>4</math> - <math>4\times4</math> squares and <math>1</math> - <math>5\times5</math> square), therefore there are <math>55-36 = 19</math> squares that contain the black square, which is <math>\boxed{\mathrm{(D)}\ 19}</math>. | ||
Revision as of 19:12, 1 February 2020
Problem
The grid shown contains a collection of squares with sizes from
to
. How many of these squares contain the black center square?
Solution
Solution 1
Since there are five types of squares: and
We must find how many of each square contain the black shaded square in the center.
If we list them, we get that
- There is
of all
squares, containing the black square
- There are
of all
squares, containing the black square
- There are
of all
squares, containing the black square
- There are
of all
squares, containing the black square
- There is
of all
squares, containing the black square
Thus, the answer is .
Solution 2
We use complementary counting. There are only and
squares that do not contain the black square. Counting, there are
-
squares, and
squares that do not contain the black square. That gives
squares that don't contain it. There are a total of
squares possible
-
squares
-
squares
-
squares
-
squares and
-
square), therefore there are
squares that 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.