2020 AMC 8 Problems/Problem 9

Revision as of 22:10, 19 November 2020 by Oceanxia (talk | contribs)

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] $\textbf{(A) }12 \qquad \textbf{(B) }16 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

its very easy to get trolled here if you dont read "no icing on the bottom"

Solution 1

Notice that all the faces with the exception of the bottom faces have the two center edge pieces with 2 faces with icing on them. This is $8\cdot 2 = 16$. Additionally, on the bottom face, the corners have 2 faces with icing, as the bottom face does not have icing. This is $4$ cubes. The total is $16+4 = 20, \textbf{(D) }20$. Note we just need to get The total is $16+4 = 20, \textbf{(D) \textbf{(D) }20$ (Error compiling LaTeX. ! File ended while scanning use of \textbf .) }20$~Windgo and minor edits made by oceanxia

==Solution 2== This is just careful casework. Consider the following diagram: https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi82Lzk1NjIyMDEyYzEwMmU2MGRhM2U3OGMzYzA0MDNmOGFmZjdkMDk3LnBuZw==&rn=YW1jIDggbm8gOS5wbmc

The face on the opposite side of the front face (hidden) is an exact copy of the front face. So the answer is$ (Error compiling LaTeX. ! Missing $ inserted.)8+4+8=\textbf{(D)}20$.

-franzliszt

Franz Liszt is in 12th grade

Video Solution

https://youtu.be/WyvmQUfxTfo

~savannahsolver

See also

2020 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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All AJHSME/AMC 8 Problems and Solutions

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