Difference between revisions of "2020 AMC 8 Problems/Problem 9"
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<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== | ==Solution 1== | ||
− | Notice that | + | 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== | ==Solution 2== | ||
− | + | 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. | |
− | + | [[File:Prob10-diagram.png|middle|center]] | |
− | + | But the hidden face is an exact copy of the front face, so the answer is <math>12+8=\boxed{\textbf{(D) }20}</math>. | |
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==Video Solution== | ==Video Solution== | ||
https://youtu.be/WyvmQUfxTfo | https://youtu.be/WyvmQUfxTfo | ||
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==See also== | ==See also== | ||
{{AMC8 box|year=2020|num-b=8|num-a=10}} | {{AMC8 box|year=2020|num-b=8|num-a=10}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 08:05, 20 November 2020
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?
Solution 1
Notice that, for a small cube which does not form part of the bottom face, it will have exactly faces with icing on them only if it is one of the
center cubes of an edge of the larger cube. There are
such edges (as we exclude the
edges of the bottom face), so this case yields
small cubes. As for the bottom face, we can see that only the
corner cubes have exactly
faces with icing, so the total is
.
Solution 2
The following diagram shows of the small cubes having exactly
faces with icing on them; that is all of them except for those on the hidden face directly opposite to the front face.
But the hidden face is an exact copy of the front face, so the answer is .
Video Solution
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.