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2020 AMC 8 Problems/Problem 9

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

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 $16+4 = 20, \textbf{(D) }20$

~Windigo

Solution 2

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

-franzliszt

Franz Liszt is in 12th grade

Video Solution

~savannahsolver

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