2020 AMC 8 Problems/Problem 9
Akash's birthday cake is in the form of a 
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 
smaller cubes, each measuring 
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]](http://latex.artofproblemsolving.com/a/8/f/a8f30d108ce27003c595273dadba04f529292b5a.png)
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 
. Additionally, on the bottom face, the corners have 2 faces with icing, as the bottom face does not have icing. This is 
cubes. The total is 
. Note we just need to get The total is $16+4 = 20, \textbf{(D) \textbf{(D) }20$ (Error compiling LaTeX. Unknown error_msg) }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. Unknown error_msg)8+4+8=\textbf{(D)}20$.
-franzliszt
Franz Liszt is in 12th grade
Video Solution
~savannahsolver
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 | ||
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