2021 Fall AMC 12B Problems/Problem 20

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 $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

Solution 1 (Direct Counting)

Divide the $2 \times 2 \times 2$ cube into two layers.

Case 1: Each layer contains 2 cubes of each color.

Note that we only need to consider the layout of the white cubes because all the other cubes will be black cubes.

There are 2 ways that the two white cubes can be arranged within each layer: adjacent or diagonal to each other.

Case 1.1: Both layers have two white cubes adjacent to each other.

Rotate the cube such that there is a white cube along the top of the first layer.

Now, consider the placement of the white cubes in the other layer:

Case 1.1.1: The white cubes are along the top.

Case 1.1.2: The white cubes are along the bottom.

This is hopefully obviously distinct from case 1.1.1.

Case 1.1.3: The white cubes are along the left.

Case 1.1.4: The white cubes are along the right.

There can be both adjacent and on top of each other, both adjacent but the two layers diagonal to each other, or both diagonal and

both diagonal but the two layers on top of each other

one adjacent and one diagonal (it doesn't matter diagonal which way), both diagonal and the two layers across from each other, In the last case, this is the same as the second case. So we have four arrangements here.

Case 2: Each layer contains 3 cubes of one color and 1 cube of the other color.

Split this cube into two layers; the sole white cube on one layer must be on the opposite corner of the sole black cube on the other layer, otherwise there will be some way to spit the cube into two layers such that there are 2 cubes of each color on each layer.

Case 3: One layer contains 0 white cubes and the other layer contains 4 white cubes.

Only 1 possible $2 \times 2 \times 2$ cube can result from this case. However, if we divide up this $2 \times 2 \times 2$ 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.

Therefore, our answer is $6 + 1 + 0 = \boxed{\textbf{(A)}\ 7}$ .

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

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2021 Fall AMC 12B Problems/Problem 20

Problem

Solution 1 (Direct Counting)

See Also