Difference between revisions of "2004 AMC 10A Problems/Problem 16"
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− | 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>, 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, therefore there are <math>55-36 = 19</math> squares that | + | 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 and so on...), therefore there are <math>55-36 = 19</math> squares that contain the black square, which is <math>\boxed{\mathrm{(D)}\ 19}</math>. |
==See also== | ==See also== |
Revision as of 12:00, 31 December 2019
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
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 and so on...), 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 |
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