Cube counting practice
Count hidden supporting cubes, or work out how many remain after removing a layer or column.
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Example
Example
Example
Count every cube, including cubes hidden behind or underneath the ones you can see.
8–12 cubes · 4 figures
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4 figures · No time limit
- Figures answered
- 0 / 4
- Active time
- 0:00
Count the hidden cubes too.
A 3D stack of cubes appears. Count every cube, including cubes hidden behind or underneath the ones you can see. Nothing floats, so every visible cube has a solid column below it.
Count by position
Viewed from above. Dots mark cubes with exactly the requested number of painted faces, including cubes hidden in the solid.
- Correct answers
- Active time
- Figures answered
- 0 / 0
Count by columns, not by faces. Every visible top has a full column underneath because nothing floats.
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Explore the method
Split a supported stack into horizontal layers and count each cube once.
On by default unless you turn it off. Saved choices are kept. Shares starts, completions, replays and error categories. For ChatGPT visits, starts and completions may include the first QuickenBrain page and country. No answers, drawings or names.
How this data is usedHelp
Each figure is a stack of identical cubes seen from one corner. Count how many cubes it contains.
Some cubes are hidden behind or below the ones you can see. They still count.
Nothing floats. Every visible cube rests on a solid column of cubes down to the ground, which lets you infer what is hidden.
Count by columns rather than by faces. Work out each column’s height, then add the heights.
In the example above, count the horizontal layers from bottom to top: 4 + 2 = 6 cubes. Count every cube once, including those hidden from view.
The five choices stay close to the true total and can differ from it by up to four cubes. There is no time limit.
Choose a button, or press 1–5 for the choice in that position. Press Enter to continue after reviewing the answer.
The cube figure is visual and cannot be completed with a screen reader alone.
This practice is for personal training and does not provide a medical or cognitive diagnosis.
Remove both marked regions. How many cubes remain?
Count regions A and B, then subtract their shared cubes only once. The stack does not fall or settle.
Name each removal before counting. Add their sizes, subtract their overlap, then remove that union from the original total.
A cube in both regions is removed once.
Hatching marks the layer or column to remove. Count every cube, including hidden supporting cubes. Subtract the removed cubes; their movement after removal does not affect the count.
All six outside faces of the large cube are painted first, including the bottom and the faces hidden from view. Then it is cut into equal small cubes. New cut faces have no paint.
A corner cube has three painted faces. An edge cube away from the corners has two. A cube in the middle of an outside face has one. A cube entirely inside has none.
“Exactly” matters. A cube with three painted faces does not count when the question asks for two. Zero can be the correct answer.
Choose one of five counts. There is no time limit or speed bonus. Finish the whole round to save it; an early end does not save a completed round.
With two cubes per edge, all eight small cubes are corners. An uncut cube, with one per edge, has six painted faces; that case is not used in the questions.
Choose a button, or press 1–5 for the choice in that position. Press Enter to continue after reviewing the answer.
Paint covers every face that touches the air but not the faces on the ground, and the hidden cubes under the top ones count too.
About this practice
Count the cubes in a 3D stack, including those hidden behind or underneath visible cubes. In Count cubes, rounds contain 3 to 7 stacks by level, scored by correct answers with no time limit.
Cube counting has five modes. Read the question first: it may ask for every cube, the cubes left after removing one or two regions, or cubes with exactly a given number of painted faces. Each mode has five answer choices and no time limit.
Count cubes: include the supports
Count cubes, not visible square faces. Every column is filled from the ground up, so a cube on top needs the cubes beneath it even when they are hidden. Work one horizontal layer at a time to avoid counting a cube twice.
The ground layer contains 4 cubes, the next contains 3, and the top contains 1. 4 + 3 + 1 = 8 cubes. The same total comes from adding the heights of the four columns: 3 + 2 + 2 + 1.
After removing cubes: subtract the whole marked part
Hatching marks either a horizontal layer or a whole column. Count every cube in that part, including hidden cubes, and subtract it from the original total. Any movement after removal does not change how many remain.
Count the original layers from bottom to top: 7 + 5 + 3 + 1 = 16. The marked column contains 3 cubes, not just its visible top. 16 − 3 = 13 cubes remain.
Combined removals: subtract shared cubes once
Two marked regions can overlap. Count the original stack, subtract region A, subtract region B, then add back their shared cubes because those were subtracted twice. A cube in both regions is removed only once. In this mode the remaining stack does not fall or settle.
Painted faces: paint first, then cut
This mode starts with one large cube painted on all six outside faces, including the bottom, before it is cut into equal small cubes. Count small cubes with exactly the requested number of painted faces. A corner has three; a cube on an edge away from corners has two; a face interior has one; a fully internal cube has none.
Use the 12 edges, leaving out each edge’s two corners. Each edge has 4 − 2 = 2 qualifying cubes. 12 × 2 = 24 cubes. Including corners would count cubes with three painted faces.
Exposed faces: check what touches the air
This mode paints the air-exposed faces of a cube stack. Faces touching another cube and bottom faces resting on the ground are unpainted. Hidden faces at the back can still be painted. For each cube, start with six faces and subtract its contacts with cubes or the ground.
Level 1. How many cubes have exactly 3 painted faces? The ground layer, labelled with each cube’s painted-face count. Rings mark matches.
The cube under the raised cube and the ground cube next to it each touch two cubes and the ground, leaving exactly 3 painted faces. The isolated ground cube and the upper cube each have five; the remaining cube has four. Answer: 2 cubes. Count qualifying cubes once each, rather than adding their painted faces.
What the levels and results mean
- Count cubes and After removing cubes: stacks grow from 6–10 cubes at level 1 to 38–57 at level 8. Removal starts with the highest layer at levels 1–2, then includes other layers and whole columns.
- Painted faces: the large cube has 2–4 small cubes along each edge at levels 1–2, increasing to 5–7 at levels 7–8. Questions can ask for exactly zero, one, two or three painted faces.
- Exposed faces: stacks become larger, and the requested face counts expand from three or four at level 1 to any count from one to five at level 8.
Each completed round has 3 to 7 questions. Your score counts correct answers, with no speed bonus. Compare the same mode and level, since counting supports and counting painted faces use different rules. Review the turned stack or layer display after an answer to locate a missed cube. An unfinished round stays in this tab, so you can continue it.
The lesson for this mode: Count cubes by layers. The research guide Spatial reasoning tests: the six question types includes cube counting among its worked examples.
For a broader mix of practices, explore the spatial practice test. Sign in free to start a test.
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On by default unless you turn it off. Saved choices are kept. Shares starts, completions, replays and error categories. For ChatGPT visits, starts and completions may include the first QuickenBrain page and country. No answers, drawings or names.
How this data is used