Cube Counting
Count cubes by how many faces are painted.
In Cube Counting, a figure is constructed by gluing cubes together. The entire figure is then painted on every exposed surface — except the bottom face resting on the table. Each question asks how many cubes have exactly a specified number of painted faces (for example, 'how many cubes have exactly two sides painted?'), including cubes hidden inside or behind the stack.
Also known as: Cubes, Painted Cubes, DAT Cube Counting
Pick an answer above, then check it.
How Cube Counting works on the DAT
- 1
Cubes are glued into a 3D figure, then painted on all exposed sides except the bottom.
- 2
You determine, for each cube, how many of its faces are painted (0–4 typically).
- 3
A question asks how many cubes have exactly N painted faces.
- 4
You must account for cubes hidden behind, under, or inside the figure.
Why students struggle
- Cubes hidden behind or beneath others that you can't directly see.
- Remembering the bottom (table) face is not painted.
- Visualising depth from a single 2D drawing.
- Keeping an accurate tally across all painted-face counts.
Common mistakes to avoid
- Forgetting cubes you can't see.
- Painting the bottom face by accident.
- Miscounting back-row or stacked cubes.
- Letting the tally drift out of sync with the total.
Proven strategies for Cube Counting
The tactics our highest-scoring students use to speed through Cube Counting without losing accuracy.
1. Build a 0–1–2–3–4 tally
Make a quick table for cubes with 0, 1, 2, 3, and 4 painted faces and fill it as you scan. One pass, one number per cube, no double counting.
2. Count total cubes first
Establish the total cube count (including hidden ones) up front. Your tally must add up to it — a built-in error check.
3. Remember the unpainted bottom
The face touching the table is never painted. Bottom-layer cubes have one fewer painted face than they appear to.
4. Hunt the hidden cubes
Cubes tucked behind or beneath the figure are usually the 0-painted-face answers. Deliberately look for them instead of counting only what's visible.
How PATCrusher helps you master Cube Counting
Rotatable 3D cube stacks
Generate unlimited cube figures and rotate them in the explanation to expose every hidden cube — difficulty modes add depth and hidden layers.
Per-cube painted-face reveal
Solutions highlight each cube and label its number of painted faces, so the full 0–4 tally is laid out for you.
Master the other PAT sections
Further reading on Cube Counting
Cube Counting — frequently asked questions
Quick answers about the DAT PAT Cube Counting section.
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