One Figure, Several Questions

Questions arrive in runs

Cube counting does not give each question its own drawing. The instructions are written for a figure that carries several questions — for each question, count how many cubes have the stated number of painted sides — and the items you will drill here reproduce that: one pile, then a run of two or three questions about it.

Here is a complete run. Watch the pile, not the words.

How many cubes have one of their exposed sides painted?

How many cubes have three of their exposed sides painted?

How many cubes have four of their exposed sides painted?

The drawing does not change across the three. It cannot: the pile is rebuilt from the figure's own identity, so every question in the run regenerates the identical picture. The only thing that moves from one question to the next is the number of painted sides being asked for.

The numbers climb, and they skip

That run asks for one, then three, then four. Two things are worth noticing.

First, a run never asks the same number twice. Three questions means three different classes of cube. If you have just answered the question about three painted sides, the next one is about something else.

Second, the numbers ascend but are not necessarily consecutive. This run skipped two entirely. Do not read the sequence as one, two, three and answer on autopilot — read each number as it comes. (That ordering is how the items in this course present a run; treat it as a habit of these drills rather than a law of the exam.)

Analyze once, answer the run

This is the section's unusual economics, and the reason cube counting rewards patience more than any other PAT section.

The expensive work is understanding the pile: how many cubes it holds, where they sit, which ones are buried. The cheap work is answering — once you know the pile, a question is a lookup. Those costs do not repeat equally. Understand the pile once and the second and third questions are nearly free; rush it, and you pay for the same mistake two or three times.

So the first question of a run is not the place to hurry. It is the place to do the whole analysis, because the run has already told you that the analysis is about to be reused. The bookkeeping that makes reuse practical — one pass that answers every number a run can ask — is the subject of the Counting Method lesson.

Knowing when the pile has changed

The run ends without announcing itself, and the next question can look deceptively familiar.

How many cubes have one of their exposed sides painted?

Compare that prompt with the first question of the run above: word for word the same. Only the drawing is different. If you track your position by the question text, you will answer this one from the previous pile's analysis and get it wrong for free.

The drawing is what tells you where you are. Glance at it before every answer; a pile you have already analyzed is instantly recognizable, and a pile you have not is instantly not. When you drill in the cube counting practice generator, each question carries a small label reading "Same figure — question 2 of 3", so you can check that instinct against the truth after you have used it.

The takeaway

One pile serves two or three questions, and every question in the run shows the identical drawing — only the stated number of painted sides changes. The run asks two or three different numbers, ascending and sometimes skipping. Do the pile once, thoroughly, and spend that work across the whole run; then check the drawing, not the wording, to know when a new pile has started.