Difference between revisions of "2021 Fall AMC 12B Problems/Problem 20"
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<math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ | <math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ | ||
10 \qquad\textbf{(E)}\ 11</math> | 10 \qquad\textbf{(E)}\ 11</math> | ||
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| + | ==Solution 1 (Direct Counting)== | ||
| + | We only need to consider the arrangement of the cubes of one color (say, white), as the four other cubes will be all of the other color (say, black). | ||
| + | |||
| + | Divide the <math>2 \times 2 \times 2</math> cube into two layers, front and back. Each layer can contain 0, 1, or 2 cubes of any given color. | ||
| + | |||
| + | Assume that we view the cube by rotating it such that there is a white cube in the upper-left of the front layer. | ||
| + | |||
| + | Case 1: First layer contains 0 white cubes. | ||
| + | |||
| + | Only 1 possible <math>2 \times 2 \times 2</math> cube can result from this case, with each layer containing all the cubes of one color. | ||
| + | |||
| + | Case 2: First layer contains 1 white cube. | ||
| + | |||
| + | |||
| + | Case 3: First layer contains 2 white cubes. | ||
==See Also== | ==See Also== | ||
| − | {{AMC12 box|year=2021 Fall|ab=B|num-a= | + | {{AMC12 box|year=2021 Fall|ab=B|num-a=21|num-b=19}} |
{{AMC10 box|year=2021 Fall|ab=B|num-a=25|num-b=23}} | {{AMC10 box|year=2021 Fall|ab=B|num-a=25|num-b=23}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 18:02, 25 November 2021
- The following problem is from both the 2021 Fall AMC 12B #20 and 2021 Fall AMC 12B #24, so both problems redirect to this page.
Problem
A cube is constructed from 
white unit cubes and black unit cubes. How many different ways are there to construct the 
cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
Solution 1 (Direct Counting)
We only need to consider the arrangement of the cubes of one color (say, white), as the four other cubes will be all of the other color (say, black).
Divide the cube into two layers, front and back. Each layer can contain 0, 1, or 2 cubes of any given color.
Assume that we view the cube by rotating it such that there is a white cube in the upper-left of the front layer.
Case 1: First layer contains 0 white cubes.
Only 1 possible cube can result from this case, with each layer containing all the cubes of one color.
Case 2: First layer contains 1 white cube.
Case 3: First layer contains 2 white cubes.
See Also
| 2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 19 |
Followed by Problem 21 |
| 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 AMC 12 Problems and Solutions | |
| 2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 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 AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
