Difference between revisions of "2020 AMC 8 Problems/Problem 9"
m |
m (→Solution 3) |
||
(16 intermediate revisions by 10 users not shown) | |||
Line 1: | Line 1: | ||
+ | ==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? | ||
Line 50: | Line 51: | ||
<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> | ||
− | |||
==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>. |
+ | Answer = D | ||
− | + | ==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>. | ||
+ | |||
+ | ==Solution 3== | ||
+ | (For Rubik's Cubers) | ||
+ | On a <math>4x4</math> rubik's cube, there are exactly <math>24</math> Hdjdb'edge' pieces, <math>8</math> 'corners', and <math>24</math> 'center' hdjdbrunr rpieces. Edge pieces have <math>2</math> frosted faces (the ones on the bottom only have one, corners have <math>3</math> frosted faces, and centers have <math>1</math>. So since we have <math>24</math> edges pieces, we minus the <math>8</math> 'edge' pieces on the bottom (they only have one frosted face), and then we add the <math>4</math> bottom 'corner' pieces (they have also 2 frosted faces). we get <math>24-8+4=\boxed{\textbf{(D) }20}</math>. | ||
− | + | -Solution by MismatchedCubing | |
− | |||
− | |||
− | + | ==Video Solution by North America Math Contest Go-Go Go== | |
− | + | https://www.youtube.com/watch?v=6LbBcFUmBr0 | |
− | + | ~North America Math Contest Go Go Go | |
− | ==Video Solution== | + | ==Video Solution by WhyMath== |
https://youtu.be/WyvmQUfxTfo | https://youtu.be/WyvmQUfxTfo | ||
~savannahsolver | ~savannahsolver | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/61c1MR9tne8 | ||
+ | |||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/YnwkBZTv5Fw?t=355 | ||
+ | |||
+ | ~Interstigation | ||
==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}} |
Latest revision as of 18:38, 16 January 2022
Contents
Problem
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 . Answer = D
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 .
Solution 3
(For Rubik's Cubers) On a rubik's cube, there are exactly Hdjdb'edge' pieces, 'corners', and 'center' hdjdbrunr rpieces. Edge pieces have frosted faces (the ones on the bottom only have one, corners have frosted faces, and centers have . So since we have edges pieces, we minus the 'edge' pieces on the bottom (they only have one frosted face), and then we add the bottom 'corner' pieces (they have also 2 frosted faces). we get .
-Solution by MismatchedCubing
Video Solution by North America Math Contest Go-Go Go
https://www.youtube.com/watch?v=6LbBcFUmBr0
~North America Math Contest Go Go Go
Video Solution by WhyMath
~savannahsolver
Video Solution
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.