Difference between revisions of "2019 AMC 8 Problems/Problem 12"
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The faces of a cube are painted in six different colors: red <math>(R)</math>, white <math>(W)</math>, green <math>(G)</math>, brown <math>(B)</math>, aqua <math>(A)</math>, and purple <math>(P)</math>. Three views of the cube are shown below. What is the color of the face opposite the aqua face? | The faces of a cube are painted in six different colors: red <math>(R)</math>, white <math>(W)</math>, green <math>(G)</math>, brown <math>(B)</math>, aqua <math>(A)</math>, and purple <math>(P)</math>. Three views of the cube are shown below. What is the color of the face opposite the aqua face? | ||
− | [[File:2019AMC8Prob12.png]] | + | <!--- [[File:2019AMC8Prob12.png]] --> |
+ | <asy> | ||
+ | unitsize(2cm); | ||
+ | pair x, y, z, trans; | ||
+ | int i; | ||
− | <math>\textbf{(A) }\text{red}\qquad\textbf{(B) }\text{white}\qquad\textbf{(C) }\text{green}\qquad\textbf{(D) }\text{brown}\qquad\textbf{(E) }\text{purple}</math> | + | x = dir(-5); |
+ | y = (0.6,0.5); | ||
+ | z = (0,1); | ||
+ | trans = (2,0); | ||
+ | |||
+ | for (i = 0; i <= 2; ++i) { | ||
+ | draw(shift(i*trans)*((0,0)--x--(x + y)--(x + y + z)--(y + z)--z--cycle)); | ||
+ | draw(shift(i*trans)*((x + z)--x)); | ||
+ | draw(shift(i*trans)*((x + z)--(x + y + z))); | ||
+ | draw(shift(i*trans)*((x + z)--z)); | ||
+ | } | ||
+ | |||
+ | label(rotate(-3)*"$R$", (x + z)/2); | ||
+ | label(rotate(-5)*slant(0.5)*"$B$", ((x + z) + (y + z))/2); | ||
+ | label(rotate(35)*slant(0.5)*"$G$", ((x + z) + (x + y))/2); | ||
+ | |||
+ | label(rotate(-3)*"$W$", (x + z)/2 + trans); | ||
+ | label(rotate(50)*slant(-1)*"$B$", ((x + z) + (y + z))/2 + trans); | ||
+ | label(rotate(35)*slant(0.5)*"$R$", ((x + z) + (x + y))/2 + trans); | ||
+ | |||
+ | label(rotate(-3)*"$P$", (x + z)/2 + 2*trans); | ||
+ | label(rotate(-5)*slant(0.5)*"$R$", ((x + z) + (y + z))/2 + 2*trans); | ||
+ | label(rotate(-85)*slant(-1)*"$G$", ((x + z) + (x + y))/2 + 2*trans); | ||
+ | </asy> | ||
+ | |||
+ | <math> \textbf{(A)} \text{ red}\qquad\textbf{(B)} \text{ white}\qquad\textbf{(C)} \text{ green}\qquad\textbf{(D)} \text{ brown}\qquad\textbf{(E)} \text{ purple} </math> | ||
==Solution 1== | ==Solution 1== | ||
− | <math>B</math> is on the top, and <math>R</math> is on the side, and <math>G</math> is on the right side. That means that (image <math>2</math>) <math>W</math> is on the left side. From the third image, you know that <math>P</math> must be on the bottom since <math>G</math> is sideways. That leaves us with the back, so the back must be <math>A</math>. The front is opposite of the back, so the answer is <math>\boxed{\textbf{(A)}\ R}</math>. | + | <math>B</math> is on the top, and <math>R</math> is on the side, and <math>G</math> is on the right side. That means that (image <math>2</math>) <math>W</math> is on the left side. From the third image, you know that <math>P</math> must be on the bottom since <math>G</math> is sideways. That leaves us with the back, so the back must be <math>A</math>. The front is opposite of the back, so the answer is <math>\boxed{\textbf{(A)}\ R}</math>. |
==Solution 2== | ==Solution 2== | ||
− | Looking closely we can see that all faces are connected with <math>R</math> except for <math>A</math>. Thus the answer is <math>\boxed{\textbf{(A)}\ R}</math>. | + | Looking closely, we can see that all faces are connected with <math>R</math> except for <math>A</math>. Thus, the answer is <math>\boxed{\textbf{(A)}\ R}</math>. |
+ | It is A, just draw it out! | ||
+ | ~phoenixfire | ||
− | ~ | + | ==Solution 3== |
+ | From pic 1 and 2, we know that the G's opposite is W. From pic 1 and 3, the B's opposite is P. So the <math>A</math>'s opposite is <math>\boxed{\textbf{(A)}\ R}</math>. | ||
+ | |||
+ | ==Video Solution by Math-X (Simple Visualization!!!)== | ||
+ | https://youtu.be/IgpayYB48C4?si=uPWa04P5Bi6wEZB-&t=3752 | ||
+ | |||
+ | ~Math-X | ||
+ | |||
+ | ==Solution Explained== | ||
+ | https://youtu.be/gOZOCFNXMhE ~ The Learning Royal | ||
==Solution 3== | ==Solution 3== | ||
Associated video - https://www.youtube.com/watch?v=K5vaX_EzjEM | Associated video - https://www.youtube.com/watch?v=K5vaX_EzjEM | ||
− | == | + | == Video Solution == |
+ | |||
+ | Solution detailing how to solve the problem: https://www.youtube.com/watch?v=VXBqE-jh2WA&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=13 | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/dru7MQO6jqs | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution (CREATIVE ANALYSIS!!!)== | ||
+ | https://youtu.be/kD4V_InGI_g | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)== | ||
+ | https://youtu.be/Xm4ZGND9WoY | ||
+ | |||
+ | ~Hayabusa1 | ||
+ | |||
+ | ==See also== | ||
{{AMC8 box|year=2019|num-b=11|num-a=13}} | {{AMC8 box|year=2019|num-b=11|num-a=13}} | ||
− |
Latest revision as of 02:22, 30 October 2024
Contents
Problem
The faces of a cube are painted in six different colors: red , white , green , brown , aqua , and purple . Three views of the cube are shown below. What is the color of the face opposite the aqua face?
Solution 1
is on the top, and is on the side, and is on the right side. That means that (image ) is on the left side. From the third image, you know that must be on the bottom since is sideways. That leaves us with the back, so the back must be . The front is opposite of the back, so the answer is .
Solution 2
Looking closely, we can see that all faces are connected with except for . Thus, the answer is .
It is A, just draw it out! ~phoenixfire
Solution 3
From pic 1 and 2, we know that the G's opposite is W. From pic 1 and 3, the B's opposite is P. So the 's opposite is .
Video Solution by Math-X (Simple Visualization!!!)
https://youtu.be/IgpayYB48C4?si=uPWa04P5Bi6wEZB-&t=3752
~Math-X
Solution Explained
https://youtu.be/gOZOCFNXMhE ~ The Learning Royal
Solution 3
Associated video - https://www.youtube.com/watch?v=K5vaX_EzjEM
Video Solution
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=VXBqE-jh2WA&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=13
Video Solution
~savannahsolver
Video Solution (CREATIVE ANALYSIS!!!)
~Education, the Study of Everything
Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)
~Hayabusa1
See also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 AJHSME/AMC 8 Problems and Solutions |