Last reviewed 22 Sept 2026 · 6 min read
A painted cube, cut up
A cube painted on all six faces and cut into smaller cubes of equal size gives:
They must total , which is the check to run before answering.
For : ✓. For : ✓.
When only some faces are painted
If only the top and bottom are painted, no small cube can have three painted faces; the counts must be rebuilt from the picture. The safe method is to think of the big cube as layers and count layer by layer, rather than to reuse a formula that assumed all six faces.
Dice
A dice has six faces in three opposite pairs. In a standard dice the numbers on opposite faces add to 7 (1-6, 2-5, 3-4); in an ordinary dice the pairs must be worked out from the views given.
- One face common to both views: the remaining four faces (two from each view) are adjacent to the common face; the face not appearing in either view is opposite to the common one.
- Two faces common to both views: the two faces that are not common are opposite to each other.
- Any two faces seen together in a single view are adjacent, never opposite — this single fact answers most questions.
A useful corollary: if a number appears in a view together with four different numbers across several views, the fifth unseen number is its opposite.