Cubes and Dice: Method with Worked Examples

Cube and dice questions test spatial reasoning, not chance. The worked examples below use three stated assumptions: every cube or die has six distinct labels, one per face; different views of the same die are consistent, so they never place opposite faces side by side; and for painted-cube counts, all six outer faces are painted before the cube is cut into equal n x n x n smaller cubes with n at least 2 (the n = 1 case is treated on its own). No probability or chance appears anywhere on this page.

Nets of a cube

A net is a flat arrangement of six squares that folds into a cube. A cube has exactly 11 distinct nets, counting rotations and reflections of a pattern as the same net. To test a candidate arrangement, fold it mentally: each of the six squares must land on a different face, with no overlaps and no gaps.

net

a flat pattern of six squares joined edge to edge that folds into a cube

opposite faces

two faces that never share an edge; a cube has 3 such pairs

adjacent faces

two faces that share an edge; each face is adjacent to exactly 4 others

Example 1: a valid net

Is the cross arrangement below a valid cube net?

    +---+
    | A |
+---+---+---+---+
| B | C | D | E |
+---+---+---+---+
    | F |
    +---+

In words: four squares B, C, D, E in a row, with A above C and F below C.

1. Fix C as the base face. Fold B and D up on its left and right edges.
2. Fold A up on C's top edge and F up on C's bottom edge; they close two more sides.
3. E, attached beyond D, folds over to close the face opposite C.
4. All six squares land on six distinct faces with no overlap, so the arrangement is a valid net.

Yes, the cross is a valid net; in fact it is one of the 11 distinct nets of a cube.

Example 2: an arrangement that is not a net

Is the 2 x 3 rectangle below a valid cube net?

+---+---+---+
| A | B | C |
+---+---+---+
| D | E | F |
+---+---+---+

In words: two rows of three squares, forming a solid rectangle.

1. Fix B as the base and fold A and C up on its left and right edges, and E up on its bottom edge.
2. D shares an edge with both A and E, and the two edges pull it to different faces: folding D down from A lands it on the same face as E, while folding it in from E lands it on the same face as A.
3. Both shared edges cannot be satisfied at once: whichever way the loop closes, two squares overlap on one face and another face is left with no square.

No. The 2 x 3 rectangle is not one of the 11 nets: its shared edges impose fold constraints that no cube can satisfy.

Opposite faces from a net

Once a net folds, opposite faces can be read off without a model. A useful shortcut: in a straight strip of four squares, the first is opposite the third and the second is opposite the fourth, because the strip wraps around the four side faces of the cube.

Example 3: find every opposite pair

The net below folds into a die. Which face is opposite which?

    +---+
    | 1 |
+---+---+---+---+
| 2 | 3 | 4 | 5 |
+---+---+---+---+
    | 6 |
    +---+

In words: a strip of four squares 2, 3, 4, 5, with 1 above 3 and 6 below 3.

1. The strip 2-3-4-5 wraps around the cube: 2 is opposite 4, and 3 is opposite 5.
2. The two remaining squares, 1 and 6, close the two open ends, so 1 is opposite 6.
3. Check: each face is adjacent to the four faces it is not opposite, and every label is used once.

Opposite pairs: 1 and 6, 2 and 4, 3 and 5.

Opposite faces from consistent views

A view of a die shows three faces meeting at one corner, so the three visible faces are always mutually adjacent. Because each face is adjacent to exactly 4 of the 6 faces, the opposite of a face is pinned down once the views together show all four of its neighbours: the one face never seen beside it is the opposite. With fewer views or only partial overlap, more than one pairing can fit, so collect views until some face's four neighbours are all visible, and check consistency as you go: no view may show a face next to its opposite.

Example 4: three views of the same die

Three views of the same die show the faces {3, 6, 4}, {5, 2, 6} and {1, 4, 5}. Find every opposite pair.

1. Face 6 appears with 3 and 4 in the first view and with 5 and 2 in the second: all four of its neighbours are 2, 3, 4, 5, so 6 is opposite the one face left, 1.
2. Face 4 appears with 3 and 6 in the first view and with 1 and 5 in the third: all four of its neighbours are 1, 3, 5, 6, so 4 is opposite 2.
3. The only unpaired faces are 3 and 5, so 3 is opposite 5.
4. Check against all three views: no derived opposite pair (1-6, 2-4 or 3-5) appears together in any view, and within each view the three faces are mutually adjacent, as they must be for one die.

Opposite pairs: 1 and 6, 2 and 4, 3 and 5.

Many standard dice are manufactured so that opposite faces sum to 7 (1-6, 2-5, 3-4). That is a convention for standard dice, not a property of every die. Use the sum-to-7 shortcut only when a problem states the die is standard; otherwise derive the opposites from a supplied net or from consistent views, as above. Notice the die in Examples 3 and 4 is not standard: 3 is opposite 5.

Rotations and rolls

To track a roll, fix the three faces you can see, top, front (facing you) and right, and update them one roll at a time. Each roll cycles four faces around the rolling axis and leaves the two side faces on that axis unchanged.

Example 5: two rolls of a labelled die

The die from Example 3 is placed with 3 on top and 6 facing you; the net then forces 4 to be on the right. It is rolled away from you once, then rolled to your right once. Which face is on top?

1. Start: top 3, front 6, right 4. The hidden faces follow from Example 3: bottom 5 (opposite 3), back 1 (opposite 6), left 2 (opposite 4).
2. Roll away from you: the front face comes to the top. Now top 6, front 5, right 4 (left and right are unchanged).
3. Roll to your right: the left face comes to the top. Now top 2, front 5, right 6.

After the two rolls, 2 is on top.

Painted cubes

Restating the model: all six outer faces of a cube are painted, then the cube is cut into equal n x n x n smaller cubes, with n at least 2. A small cube's painted faces equal the number of outer faces it touches, so position decides the count: corner cubes touch 3 faces, edge cubes touch 2, face-centre cubes touch 1 and interior cubes touch 0.

corner cubes, 3 painted faces

always 8 of them, for any n >= 2

edge cubes, 2 painted faces

12(n - 2): each of the 12 edges holds n - 2 cubes between its corners

face-centre cubes, 1 painted face

6(n - 2)^2: an (n - 2) x (n - 2) patch on each of the 6 faces

interior cubes, 0 painted faces

(n - 2)^3: the untouched cube inside

Example 6: a 3 x 3 x 3 painted cube

A cube is painted on all six faces and cut into 3 x 3 x 3 = 27 equal smaller cubes. How many small cubes have 3, 2, 1 and 0 painted faces?

1. Three painted faces, corners: 8.
2. Two painted faces, edges: 12 x (3 - 2) = 12.
3. One painted face, face centres: 6 x (3 - 2)^2 = 6.
4. Zero painted faces, interior: (3 - 2)^3 = 1, the single cube at the very centre.
5. Check the total: 8 + 12 + 6 + 1 = 27 = 3^3, so every small cube is accounted for.

8 cubes with 3 painted faces, 12 with 2, 6 with 1, and 1 with none.

For n = 2 the formulas give 8 corner cubes and nothing else, matching the 8 cubes of a 2 x 2 x 2 cut, each painted on 3 faces. The case n = 1 is different in kind: the single uncut cube is painted on all 6 faces, and the 0-to-3-face scheme above does not apply to it.

Summary

Nets

11 distinct nets; a candidate is valid only if all six squares fold to six distinct faces

Strip shortcut

in a straight strip of four squares, 1st is opposite 3rd and 2nd is opposite 4th

Views

check consistency first; once all four neighbours of a face are visible, the missing face is its opposite

Standard dice

opposite faces sum to 7 by convention only; derive opposites unless the problem says standard

Rolls

track top, front, right one roll at a time; side faces on the rolling axis stay fixed

Painted n-cube

corners 8, edges 12(n - 2), faces 6(n - 2)^2, interior (n - 2)^3, total n^3, for n >= 2; n = 1 is separate

Frequently asked questions

How many distinct nets does a cube have?

Eleven, counting rotations and reflections of a pattern as the same net. A candidate arrangement is a net only if it folds so all six squares become six distinct faces, with no overlaps or gaps.

Do the opposite faces of every die add up to seven?

No. Opposite faces summing to seven is a manufacturing convention for standard dice, not a rule for every die. Use it only when the problem says the die is standard; otherwise derive opposites from a net or consistent views.

How do two views of the same die help find opposite faces?

Each view shows three mutually adjacent faces. Once the views together show all four neighbours of a face, the one face never seen beside it is its opposite; with less information, more than one pairing can fit.

What is assumed before counting painted faces on a cut cube?

That all six outer faces are painted before cutting and the cube is divided into equal n x n x n smaller cubes with n at least 2. An uncut single cube, n = 1, has 6 painted faces and sits outside the 0-to-3-face scheme.

Can two views of the same die disagree?

Not if they are accurate views of the same labelled die. Consistent views never place opposite faces side by side and preserve each shared face's cyclic neighbour order; a conflict means the views or labels need checking.

Sources and scope

Net count and the standard-die convention checked against Wikipedia, Net (polyhedron) and Dice, retrieved 15 November 2026; the Olympiad tie is the SOF Class 9 Olympiad reasoning workbook, which lists Cubes and Dice as chapter 22, at sofworld.org, retrieved 15 November 2026. That tie is Olympiad enrichment, not a board-syllabus claim. This page covers deterministic spatial reasoning only: no dice probability, and no surface-area or volume numericals.

Related reading on this site: introduction to probability (class 9), probability (class 10), and surface areas and volumes (class 10).