Difference between revisions of "2021 Fall AMC 12B Problems/Problem 20"
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Assume that we view the cube by rotating it such that there is a white cube in the upper-left of the front layer. | 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 | + | Case 1: First layer contains 2 white cubes. |
| − | |||
| − | |||
Case 2: First layer contains 1 white cube. | Case 2: First layer contains 1 white cube. | ||
| + | Case 3: 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. However, if we divide up this <math>2 \times 2 \times 2</math> cube in the direction perpendicular to the way we first did, then we see this is a repeat of the case where we have 2 white cubes in the first layer and the second layer is arranged such that whites and blacks are straight with each other. | |
==See Also== | ==See Also== | ||
Revision as of 19:06, 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 2 white cubes.
Case 2: First layer contains 1 white cube.
Case 3: 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. However, if we divide up this cube in the direction perpendicular to the way we first did, then we see this is a repeat of the case where we have 2 white cubes in the first layer and the second layer is arranged such that whites and blacks are straight with each other.
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. 
