Ada has a set of identical cubes. She makes solids by gluing together 4 of these cubes. When cube faces are glued together, they must coincide. Each of the 4 cubes must have a face that coincides with a face of at least one of the other 3 cubes. One such solid is shown. The number of unique solids that Ada can make using 4 cubes is
(A) 5
(B) 6
(C) 7
(D) 8
(E) 10

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-053.jpg?height=263&width=358&top_left_y=1403&top_left_x=1339)
When Ada glues cube faces together, they must coincide.

Also, each of the 4 cubes must have a face that coincides with a face of at least one of the other 3 cubes.

We this in mind, we begin by repositioning the 4 identical cubes, attempting to construct figures different from those already constructed.

Figure 1 shown below was given in the question.

The next 4 figures (numbered 2 to 5 ) have the same thickness as figure 1.

That is, each of the first 5 figures is 1 cube thick from front to back.

These are the only 5 figures that can be constructed having this 1 cube thickness and these 5 figures are all unique.

Can you verify this for yourself? Attempt to construct a different figure with a 1 cube thickness. Now rotate this figure to see if it will match one of the first 5 figures shown.

Next, we consider constructing figures with a front to back depth of 2 cubes.

The only 3 unique figures that can be constructed in this way are labeled 6 to 8 below.

While figures 7 and 8 look to be the same, there is no way to rotate one of them so that it is identical to the other.

Thus, not only are these 3 figures the only figures that can be constructed with a 2 cube thickness, but they are all different from one another.

Since no rotation of any of these 3 figures will result in it having a 1 cube thickness from front to back, the figures 6 to 8 are all different than any of the previous 5 figures.

With some rotation, we can verify that the only figures having a 3 cube thickness have already been constructed (figures 1,2 and 5).

Similarly, the only figure that can possibly be built with a 4 cube thickness has already been constructed (figure 4).

The 8 figures shown below are the only figures that Ada can construct.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-253.jpg?height=488&width=1133&top_left_y=2030&top_left_x=534)

ANSWER: (D)