Difference between revisions of "2012 AMC 10A Problems/Problem 14"
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In total, there are <math>15 \cdot 30 + 15 + 15 + 1 = 481</math> tiles, giving an answer of <math>\boxed{\textbf{(B)}\ 481}</math> | In total, there are <math>15 \cdot 30 + 15 + 15 + 1 = 481</math> tiles, giving an answer of <math>\boxed{\textbf{(B)}\ 481}</math> | ||
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| + | ==Solution 3== | ||
| + | |||
| + | For every <math>2 \times 31</math> strip, there are 31 black tiles and 31 white tiles. There are 15 <math>2 \times 31</math> strips on checkerboar. | ||
| + | |||
| + | The last <math>1 \times 31</math> strip starts with black tile, so it has 16 black tiles. | ||
| + | |||
| + | In total, there are <math>31\times 15+16=481</math> black tiles. | ||
| + | |||
| + | ~Bran_Qin | ||
== See Also == | == See Also == | ||
Revision as of 22:24, 7 August 2021
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 
Note: When solving 
+
, you only need to calculate the units digit.
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 
Solution 3
For every 
strip, there are 31 black tiles and 31 white tiles. There are 15 strips on checkerboar.
The last 
strip starts with black tile, so it has 16 black tiles.
In total, there are 
black tiles.
~Bran_Qin
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
