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Cube Counting (CC) Questions

What is cube counting?

Cube counting problems are the second to last group of problems that appear on the perceptual ability test (PAT) of the dental admissions test (DAT) and comprise Items 61-75. In these problems, you will be shown a figure created by cementing together identically sized cubes. You are to imagine that, after the cubes were cemented, they were placed on the ground and then painted (note that the bottom of the structure is NOT painted). Your task is to determine how many cubes have a certain number of exposed sides painted. Specifically, you may be asked:

  • How many cubes have one of their sides painted?
  • How many cubes have two of their sides painted?
  • How many cubes have three of their sides painted?
  • How many cubes have four of their sides painted
  • How many cubes have five of their sides painted?
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Here are the rules:

Being asked how many cubes have x “number of sides painted” is the same as being asked how many cubes have x number of exposed sides in the structure. All exposed sides are painted. (But the bottom of the structure is not exposed and is not painted.)


The only hidden cubes are those required to support the other cubes in the structure. For instance, you know that there is one invisible cube in the back of the structure here, because it is needed to support the cube on top of it. (This invisible cube must have two exposed sides, or two sides painted.)

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There are no problems for which zero (0) is a correct answer.


Sometimes on the DAT, you may encounter a structure with a “floating cube, like the one marked in pink here.” These will only be present if you can SEE them. You still have to assume the presence of supporting cubes in areas you cannot see.

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Note: Directions given on test day may be formatted differently, but objectives are the same.

Example problem

Here is a typical cube counting problem you’d encounter on your DAT, along with an Erudition-style explanation to help you understand how to solve it.

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The key here shows how many cubes have each number of sides painted. Notice that there are three cubes that are completely hidden from the front view. As shown, there are six cubes that have one exposed side.

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Breakdown of correct answer:

There are three cubes in the front part of the structure that have one exposed side each. Notice that all other sides of these cubes are covered by other cubes.

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There are additional three cubes in the back part of the structure that have one exposed side each. These cubes are hidden from the front view, but notice their position when the structure is turned around.

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CC tips & strategies!

When you get to a new cube counting structure on your exam, make sure you take the time to understand the structure before answering the associated problems. Account for the cubes in each row and at each vertical level.

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Watch our course video where we go over over three types of cube counting strategies.

Strategies range from the “super structured” approach, which involves accounting for all cubes before going through the problems, to the “keep things moving” approach, where you answer questions independently.

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When counting cubes using any approach, move through the structure in an orderly way.

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Use our course CC videos for other unique tips on approaching and analyzing cube structures.

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Also check out PAT videos we’ve shared on YouTube!

Check out our free YouTube videos or upgrade your account on to access our full course videos!

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