The Item Format and Official Rules

What a cube counting item looks like

Every cube counting item is built on one drawing: a pile of same-size cubes, cemented together and drawn from a single fixed viewpoint. Beneath it sits a question that names one exact number of painted sides and asks how many cubes in the pile have that number, followed by five choices.

How many cubes have two of their exposed sides painted?

A

1 cube

B

2 cubes

C

3 cubes

D

4 cubes

E

5 cubes

Cube counting is questions 61 through 75 of the Perceptual Ability Test, the fifth of its six sections. One drawing normally carries more than one question — the next topic is about what that buys you.

The four rules that define the pile

The instructions tell you everything you are entitled to assume about the pile. Four sentences do all the work:

1. The figure is made by cementing together cubes of the same size. No cube is bigger than another, so painted counts are comparable everywhere in the pile.
2. After being cemented together, the group was painted on all sides except for the bottom on which it is resting. The paint went onto the assembled pile, not onto loose cubes.
3. Faces where two cubes are cemented together are not painted. The brush never reached inside a joint.
4. The only hidden cubes are those required to support other cubes. Nothing is tucked out of sight for decoration.

Rules 2 and 3 are the entire scoring system. A face carries paint when it faced open air at the moment the pile was painted, and nothing else does.

What the rules do to one cube

Here is a four-cube pile with each drawn cube's painted-face count written on it.

The cube on top shows 5. Its top face and all four of its sides met open air; only its underside is cemented to the cube beneath it, and that joint took no paint. The two cubes on the ground show 4 each: a painted top, three painted sides, one side cemented to a neighbor, and an underside sitting on the table.

That is the pattern to internalize. In these piles a cube's underside is either resting on the table or cemented to the cube below it, so it never counts as painted — the two cases the rules exclude are the only two that occur. A cube can therefore reach five painted faces, never six.

Hidden cubes hold things up

Count the cubes in that figure and you find three. The pile contains four. The cube on top cannot float, so a fourth cube sits underneath it, wedged between the two you can see — invisible because the drawing shows only a cube's top face and the two faces that turn toward you, and all three of that cube's are covered.

It carries no badge because the drawing gives it no face to write on. Rule 4 is what makes this safe rather than guesswork: the only hidden cubes are the ones required to hold other cubes up. You never have to wonder whether a decorative cube is stashed somewhere out of view. If a cube is not needed as support, it is not there. Locating those support cubes is a skill of its own, covered in the Reading the Pile lesson.

The question names an exact number

A question asks for cubes with a stated number of painted sides, and it means exactly that number — not that number or more.

In this pile, a question about four painted sides selects the two tinted cubes and stops there. The cube on top does not qualify: it has five, which is not four. Read the question's number as a precise class, and you will never be tempted to sweep in the cubes just above or below it.

The takeaway

An item is one pile of cemented same-size cubes plus a question naming an exact number of painted sides. Paint covers every outside surface except the bottom the pile rests on, and cemented joints are never painted, so in these piles a cube's underside never counts. Any cube you cannot see is there to support another cube, and the number in the question is an exact class, not a floor.