Cube and Dice

The build of a cube and the relation of faces — adjacent and opposite.

Every face has four neighbours (sharing its four edges) and one face opposite that never touches it. So any two faces are either neighbours or opposite — there is no third relation.

• At one corner of a cube three faces meet, and the three are neighbours of one another.
• So at most three faces can be seen at once, and the three seen are never opposite faces.
• The three hidden faces are exactly the opposites of the three visible ones.

What a net is, the eleven nets of a cube and the rules to test whether a figure is a net.

Cut the six faces of a cube apart so that they stay joined along edges and spread them flat — that gives a net. There are 35 shapes made by joining six squares, but only 11 of them fold into a cube (turns and flips counted as the same).

The eleven nets above: with a row of four squares (the other two squares one above and one below the row, six ways) · two-three-one (three ways) · the two-two-two staircase (one) · three-three (one). You need not memorise them; three rules below are enough:

1. No square block of four — if four squares form a $2\times 2$ block it cannot make a cube (three faces meet at a corner, not four).
2. No five in a row — five squares in one line make the fifth land on the first.
3. If there is a row of four the other two squares must be on opposite sides of the row (one above, one below); on the same side they would land on the same face.

Rules to spot opposite faces from a net without folding.

On folding, the squares of the net become the faces of the cube. Two squares sharing an edge in the net remain neighbours after folding, but some squares far apart in the net land opposite each other. Two ways to spot them:

1. The skip-one rule — in a straight line of three squares (or a row of four) the first and the third squares are opposite; in a row of four, first–third and second–fourth.
2. Take the neighbours away — count the four neighbours of the face whose opposite you want; of the five other letters, the one left over is its opposite.

Choosing the right cube of four when letters are written on a net.

When the faces of a net carry letters and you are asked which cube it folds into, two checks are enough:

1. The opposite pairs — find from the net which two letters are opposite. A cube picture showing both together is wrong.
2. The order check — if three letters at a corner are in the wrong order (a mirror image) that cube cannot be made either, even though the letters are right. To check the order find a corner of three letters in the net and see in which clockwise order they meet.

The rules of a standard die and the numbers on hidden faces.

Finding the opposite number from the common number and the list of neighbours.

The top or bottom number at the end after rolling a standard die several times.

The four classes of small cubes of an $n\times n\times n$ cube and their formulas; the same for a cuboid.

How the counts of the four classes change when only some faces are painted.

Counting cuts and pieces when a cube is sliced.

Which tool to pick up on seeing the kind of question.

The commonest reasons for losing marks on cubes and dice.

Simple nets, the standard die and small cubes.

Cubes from nets, three positions, rolling and partly painted cubes.

Hard nets, three positions, five rolls and big painted cubes.