# Cubes and Cubical Dice Tests

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Six-sided dice are cube-shaped.

A cuboid is a solid figure bounded by six faces, forming a convex polyhedron.

Important points:

1. In a cube or a cuboid there are six faces each.
2. In a cube length, breadth and height are same while in cuboid these are different.
3. In a cube the number of unit cubes = (side) 3 while in cuboid the number of unit cube = (l  b  h).

In this type of questions a stack of cubes is given and the different sides of the stack are painted with different colours. Questions are then given asking how many cubes have no paint on them, how many have just one colour on them, how many have two colours etc.

Short-cut method

For a standard stack of cubes having no irregularity and the colours painted evenly on all the surfaces, we have a much simpler mathematical formula.

Suppose we called the stack as a 4 × 4× 4 stack since its every edge is made up of 4 small cubes. Now, for an n ×  n ×  n stack of cubes, with three colours painted on them and same colours painted on opposite faces, we have:

The number of cubes having 3 surfaces painted is always 8.

1. The number of cubes having 2 surfaces painted is equal to 12 (n - 2).
2. The number of cubes having only 1 surface painted is equal to 6(n - 2)2
3. The number of cubes having no surfaces painted is equal to (n - 2)2

Now, in the example above we have a 4  × 4  × 4 stack, hence, n = 4. Therefore,

1. Number of cubes having 3 surfaces painted is 8.
2. Number of cubes having 2 surface painted is 12 (n - 2) = 12 ×  (4 - 2) = 24.
3. Number of cubes having only 1 surface painted is 6(n - 2)3  = 6 ×  (4 - 2)2 = 24.
4. Number of cubes having no surfaces painted is (n - 2)3 = 23 = 8.

The formula given above works only for a cubical stack i.e. a stack, which itself looks like a cube. We can have cuboidal stacks too, where the stack, although made up of small cubes, looks like a cuboid. For example we may have an m× n×  p stack where the three edges of the stack have been made of m cubes, n cubes, and p cubes respectively. For example a  4×  5 × 6  cuboidal stack is given.

The cuboid is made of 4×  5 × 6 = 120 small cubical dice. Now if such an m× n× p cuboidal stack is painted with three colours with same colour on opposite faces, we have the following:

1. Number of cubes having three surfaces painted is always equal to 8.
2. Number of cubes having two surfaces painted is equal to 4 [(m-2) + (n-2) + (p-2)].
3. Number of cubes having only one surface painted is equal to 2 [(m-2) (n-2) + (n-2) (p-2) + (p-2) (m-2)].
4. Number of cubes having no surfaces painted is equal to (m-2) (n-2) (p- 2).