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Difference between revisions of "2023 AMC 8 Problems/Problem 17"

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<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
 
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
  
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==Solution (Intuition)==
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==Solution 1==
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The answer is <math>\boxed{\textbf{(A)}\ 1}.</math> Use intuition to bring it down to <math>2</math> guesses <math>1</math> or <math>2</math> and guess from there or you could actually fold the paper.
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In the net, face <math>5</math> shares a vertex with the bottom-left corner of face <math>Q</math>, but it doesn't share an edge with face <math>Q</math>, so face <math>5</math> must be the face diagonally across from face <math>Q</math>'s bottom-left corner in the octahedron. Notice that in the octahedron, this face is the only one that doesn't share any vertices with face <math>?</math>. On the net, face <math>5</math> shares at least 1 vertex with all of the other faces except for face <math>1</math>. Thus, the number of face we are being asked for is <math>\boxed{\textbf{(A)}\ 1}</math>.  
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-apex304
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~UnknownMonkey, apex304, MRENTHUSIASM
  
 
==Solution 2==
 
==Solution 2==
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In the net, face <math>5</math> shares a vertex with the bottom-left corner of face <math>Q</math>, but it doesn't share an edge with face <math>Q</math>, so face <math>5</math> must be the face diagonally across from face <math>Q</math>'s bottom-left corner in the octahedron. Notice that in the octahedron, this face is the only one that doesn't share any vertices with face <math>?</math>. On the net, face <math>5</math> shares at least 1 vertex with all of the other faces except for face <math>1</math>. Thus, the number of face we are being asked for is <math>\boxed{\textbf{(A)}\ 1}</math>. -UnknownMonkey -minor edits apex304
 
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==Solution 3==
 
 
We label the octohedron going triangle by triangle until we reach the <math>?</math> triangle. The triangle to the left of the <math>Q</math> should be labeled with a <math>6</math>. Underneath triangle <math>6</math> is triangle <math>5</math>. The triangle to the right of triangle <math>5</math> is triangle <math>4</math> and further to the right is triangle <math>3</math>. Finally, the side of triangle <math>3</math> under triangle <math>Q</math> is <math>2</math>, so the triangle to the right of <math>Q</math> is <math>\boxed{\textbf{(A)}\ 1}</math>.  
 
We label the octohedron going triangle by triangle until we reach the <math>?</math> triangle. The triangle to the left of the <math>Q</math> should be labeled with a <math>6</math>. Underneath triangle <math>6</math> is triangle <math>5</math>. The triangle to the right of triangle <math>5</math> is triangle <math>4</math> and further to the right is triangle <math>3</math>. Finally, the side of triangle <math>3</math> under triangle <math>Q</math> is <math>2</math>, so the triangle to the right of <math>Q</math> is <math>\boxed{\textbf{(A)}\ 1}</math>.  
  

Revision as of 15:27, 26 January 2023

Problem

A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$?

[asy] // Diagram by TheMathGuyd import graph; // The Solid // To save processing time, do not use three (dimensions) // Project (roughly) to two size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05)); pair g = (-8,0); // Define Gap transform real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow // Time for the NET pair DA,DB,DC,CD,O; DA = (6.92820323028,0); DB = (3.46410161514,6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label("$7$",trf*DC); label("$Q$",trf*DC+DA-DB); label("$5$",trf*DC-DB); label("$3$",trf*DC-DA-DB); label("$6$",trf*CD); label("$4$",trf*CD-DA); label("$2$",trf*CD-DA-DB); label("$1$",trf*CD-2DA); [/asy]

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Solution 1

In the net, face $5$ shares a vertex with the bottom-left corner of face $Q$, but it doesn't share an edge with face $Q$, so face $5$ must be the face diagonally across from face $Q$'s bottom-left corner in the octahedron. Notice that in the octahedron, this face is the only one that doesn't share any vertices with face $?$. On the net, face $5$ shares at least 1 vertex with all of the other faces except for face $1$. Thus, the number of face we are being asked for is $\boxed{\textbf{(A)}\ 1}$.

~UnknownMonkey, apex304, MRENTHUSIASM

Solution 2

We label the octohedron going triangle by triangle until we reach the $?$ triangle. The triangle to the left of the $Q$ should be labeled with a $6$. Underneath triangle $6$ is triangle $5$. The triangle to the right of triangle $5$ is triangle $4$ and further to the right is triangle $3$. Finally, the side of triangle $3$ under triangle $Q$ is $2$, so the triangle to the right of $Q$ is $\boxed{\textbf{(A)}\ 1}$.

~hdanger

Animated Video Solution

https://youtu.be/ECqljkDeA5o

~Star League (https://starleague.us)

Video Solution by OmegaLearn (Using 3D Visualization)

https://youtu.be/gIjhiw1CUgY

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=3789

See Also

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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

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