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Difference between revisions of "2020 AMC 8 Problems/Problem 9"

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==Problem==
 
Akash's birthday cake is in the form of a <math>4 \times 4 \times 4</math> inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into <math>64</math> smaller cubes, each measuring <math>1 \times 1 \times 1</math> inch, as shown below. How many of the small pieces will have icing on exactly two sides?
 
Akash's birthday cake is in the form of a <math>4 \times 4 \times 4</math> inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into <math>64</math> smaller cubes, each measuring <math>1 \times 1 \times 1</math> inch, as shown below. How many of the small pieces will have icing on exactly two sides?
  
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</asy>
 
</asy>
 
<math>\textbf{(A) }12 \qquad \textbf{(B) }16 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24</math>
 
<math>\textbf{(A) }12 \qquad \textbf{(B) }16 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24</math>
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==Solution 1==
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Notice that, for a small cube which does not form part of the bottom face, it will have exactly <math>2</math> faces with icing on them only if it is one of the <math>2</math> center cubes of an edge of the larger cube. There are <math>12-4 = 8</math> such edges (as we exclude the <math>4</math> edges of the bottom face), so this case yields <math>2 \cdot 8 = 16</math> small cubes. As for the bottom face, we can see that only the <math>4</math> corner cubes have exactly <math>2</math> faces with icing, so the total is <math>16+4 = \boxed{\textbf{(D) }20}</math>.
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==Solution 2==
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The following diagram shows <math>12</math> of the small cubes having exactly <math>2</math> faces with icing on them; that is all of them except for those on the hidden face directly opposite to the front face.
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[[File:Prob10-diagram.png|middle|center]]
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But the hidden face is an exact copy of the front face, so the answer is <math>12+8=\boxed{\textbf{(D) }20}</math>.
 +
 +
==Video Solution by WhyMath==
 +
https://youtu.be/WyvmQUfxTfo
 +
 +
~savannahsolver
 +
 +
==Video Solution==
 +
https://youtu.be/61c1MR9tne8
 +
 +
==Video Solution by Interstigation==
 +
https://youtu.be/YnwkBZTv5Fw?t=355
 +
 +
~Interstigation
 +
 +
==See also==
 +
{{AMC8 box|year=2020|num-b=8|num-a=10}}
 +
{{MAA Notice}}

Revision as of 14:30, 18 April 2021

Problem

Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?

[asy] /* Created by SirCalcsALot and sonone Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote */ import three; currentprojection=orthographic(1.75,7,2); //++++ edit colors, names are self-explainatory ++++ //pen top=rgb(27/255, 135/255, 212/255); //pen right=rgb(254/255,245/255,182/255); //pen left=rgb(153/255,200/255,99/255); pen top = rgb(170/255, 170/255, 170/255); pen left = rgb(81/255, 81/255, 81/255); pen right = rgb(165/255, 165/255, 165/255); pen edges=black; int max_side = 4; //+++++++++++++++++++++++++++++++++++++++ path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle; path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle; path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle; for(int i=0; i<max_side; ++i){ for(int j=0; j<max_side; ++j){ draw(shift(i,j,-1)*surface(topface),top); draw(shift(i,j,-1)*topface,edges); draw(shift(i,-1,j)*surface(rightface),right); draw(shift(i,-1,j)*rightface,edges); draw(shift(-1,j,i)*surface(leftface),left); draw(shift(-1,j,i)*leftface,edges); } } picture CUBE; draw(CUBE,surface(leftface),left,nolight); draw(CUBE,surface(rightface),right,nolight); draw(CUBE,surface(topface),top,nolight); draw(CUBE,topface,edges); draw(CUBE,leftface,edges); draw(CUBE,rightface,edges); // begin made by SirCalcsALot int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}}; for (int i = 0; i < max_side; ++i) { for (int j = 0; j < max_side; ++j) { for (int k = 0; k < min(heights[i][j], max_side); ++k) { add(shift(i,j,k)*CUBE); } } } [/asy] $\textbf{(A) }12 \qquad \textbf{(B) }16 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

Solution 1

Notice that, for a small cube which does not form part of the bottom face, it will have exactly $2$ faces with icing on them only if it is one of the $2$ center cubes of an edge of the larger cube. There are $12-4 = 8$ such edges (as we exclude the $4$ edges of the bottom face), so this case yields $2 \cdot 8 = 16$ small cubes. As for the bottom face, we can see that only the $4$ corner cubes have exactly $2$ faces with icing, so the total is $16+4 = \boxed{\textbf{(D) }20}$.

Solution 2

The following diagram shows $12$ of the small cubes having exactly $2$ faces with icing on them; that is all of them except for those on the hidden face directly opposite to the front face.

Prob10-diagram.png

But the hidden face is an exact copy of the front face, so the answer is $12+8=\boxed{\textbf{(D) }20}$.

Video Solution by WhyMath

https://youtu.be/WyvmQUfxTfo

~savannahsolver

Video Solution

https://youtu.be/61c1MR9tne8

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=355

~Interstigation

See also

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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