Difference between revisions of "2012 AMC 10A Problems/Problem 14"
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<math> \textbf{(A)}\ 480 \qquad\textbf{(B)}\ 481 \qquad\textbf{(C)}\ 482 \qquad\textbf{(D)}\ 483 \qquad\textbf{(E)}\ 484</math> | <math> \textbf{(A)}\ 480 \qquad\textbf{(B)}\ 481 \qquad\textbf{(C)}\ 482 \qquad\textbf{(D)}\ 483 \qquad\textbf{(E)}\ 484</math> | ||
− | == Solution == | + | == Solution 1== |
+ | There are 15 rows with 15 black tiles, and 16 rows with 16 black tiles, so the answer is <math>15^2+16^2 =225+256= \boxed{\textbf{(B)}\ 481}</math> | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | We build the <math>31 \times 31</math> checkerboard starting with a board of <math>30 \times 30</math> that is exactly half black. There are <math>15 \cdot 30</math> black tiles in this region. | ||
+ | |||
+ | Add to this <math>30 \times 30</math> checkerboard a <math>1 \times 30</math> strip on the bottom that has <math>15</math> black tiles. | ||
− | + | Add to this <math>31 \times 30</math> checkerboard a <math>31 \times 1</math> strip on the right that has <math>15 + 1</math> black tiles. | |
+ | |||
+ | In total, there are <math>15 \cdot 30 + 15 + 15 + 1 = 481</math> tiles, giving an answer of <math>\boxed{\textbf{(B)}\ 481}</math> | ||
== See Also == | == See Also == | ||
{{AMC10 box|year=2012|ab=A|num-b=13|num-a=15}} | {{AMC10 box|year=2012|ab=A|num-b=13|num-a=15}} |
Revision as of 00:31, 9 February 2012
Contents
Problem
Chubby makes nonstandard checkerboards that have squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
Solution 1
There are 15 rows with 15 black tiles, and 16 rows with 16 black tiles, so the answer is
Solution 2
We build the checkerboard starting with a board of that is exactly half black. There are black tiles in this region.
Add to this checkerboard a strip on the bottom that has black tiles.
Add to this checkerboard a strip on the right that has black tiles.
In total, there are tiles, giving an answer of
See Also
2012 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 |