Difference between revisions of "2023 AMC 10A Problems/Problem 20"
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+ | ==Video Solution by OmegaLearn == | ||
+ | https://youtu.be/Zrqy5yGYlvQ | ||
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==See Also== | ==See Also== | ||
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Revision as of 23:43, 9 November 2023
Contents
Problem
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
Let a "tile" denote a square and "square" refer to
.
We first have possible ways to fill out the top left square. We then fill out the bottom right tile. 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
Solution 2
Note that there can be no overlap between colors in each square.
Then, we can choose color to be in the center.
Now, we have some casework:
Case 1: 1 color is placed in 4 corners and then others are placed on opposite edges.
There's
ways to do this.
Case 2: 2 colors are placed with 2 in adjacent corners and 1 edge opposite them. The final color is placed in the remaining 2 edges.
The orientation of the 2 colors has 2 possibilities, and there are
color permutations.
There can't be any more ways to do this, as we have combined all cases such that each color is used once and only once per square.
We multiply the start with the sum of the 2 cases:
.
Solution 3
Let’s call the top-right corner color A, the top-middle color B, the top-right color C, and so on, with color D being the middle row, and right corner square, and color G being the bottom-left square’s color. WLOG A=Red, B=White, D=Blue, and E=Green. We will now consider squares C and F’s colors. Case 1 : C=Red and F=Blue In this case, we get that G and H have to be Red and White in some order, and the same for H and I. We can color this in 2 ways. Case 2 : C=Blue and F=Red In this case, one of G and H needs to be White and Red, and H and I needs to be White and Blue. There is 1 way to color this. In total, we get 24*(2+1)=72 ways to color the grid. Our answer is D.
-paixiao
Video Solution 1
https://www.youtube.com/watch?v=VWZBpT9lt0Q&t=6s
-paixiao
Video Solution by OmegaLearn
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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.