Difference between revisions of "2023 AMC 12A Problems/Problem 20"
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| − | + | Each square in a <math>3\times3</math> grid of squares is colored red, white, blue, or green so that every <math>2\times2</math> square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible? | |
| − | + | ==Solution 1== | |
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| + | We first have <math>4!=24</math> possible ways to fill out the top left square. We then fill out the bottom right tile (let a "tile" denote a <math>1\times1</math> square and "square" refer to <math>2\times2</math>). In the bottom right square, we already have one corner filled out (from our initial coloring), and we now have <math>3</math> options left to pick from. | ||
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| + | We then look at the right middle tile. It is part of two squares: the top right and top left. Among these squares, <math>3</math> colors have already been used, so we only have one more option for it. Similarly, every other square only has one more option, so we have a total of <math>3\cdot4!=72</math> ways. | ||
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| + | ~Technodoggo | ||
Revision as of 21:17, 9 November 2023
Each square in a 
grid of squares is colored red, white, blue, or green so that every 
square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
Solution 1
We first have 
possible ways to fill out the top left square. We then fill out the bottom right tile (let a "tile" denote a 
square and "square" refer to ). In the bottom right square, we already have one corner filled out (from our initial coloring), and we now have 
options left to pick from.
We then look at the right middle tile. It is part of two squares: the top right and top left. Among these squares, colors have already been used, so we only have one more option for it. Similarly, every other square only has one more option, so we have a total of 
ways.
~Technodoggo
