Cubes and Dice

About Cubes and Dice

Two kinds of question are set under this heading: a painted block that is cut into small cubes, and a dice shown in three positions. In the first, whether a small cube is painted depends only on where it sits in the block. In the second, two faces that meet along an edge can never be opposite, so finding which numbers touch a face is what settles the answer.

What you need to understand

A small cube has one painted face for every side of the block it lies on: two ends in a direction means that direction is painted, and a cube that is not at either end is not painted in that direction.

That gives four kinds of small cube - the corners with three painted faces, the ones on an edge between two corners with two, the ones in the middle of a face with one, and the ones inside with none.

For a block of length l, breadth b and height h cut into unit cubes: 8 cubes have three painted faces, 4((l-2)+(b-2)+(h-2)) have two, 2((l-2)(b-2) + (b-2)(h-2) + (l-2)(h-2)) have one, and (l-2)(b-2)(h-2) have none.

The number of corners is always 8, whatever the size of the block. The other three counts grow with the size.

A view of a dice shows three faces meeting at one corner, so those three faces are neighbours of each other and no two of them can be opposite.

Every face has exactly four neighbours, and the fifth face - the one not touching it - is the opposite face. So if you can account for all four numbers touching a face, the number left over is opposite it.

A dice and its mirror image are two different dice, even though the same three numbers meet at a corner. Which of the two a set of views shows is what fixes the face opposite another.

How to work through these questions

For a painted block, note the three dimensions first, then count the four kinds of cube in order: corners, then edge cubes, then face cubes, then the ones inside.

The cubes with no paint are the ones that survive after a layer one cube thick is taken off every side, so they form a block of (l-2) by (b-2) by (h-2).

Check the four counts add up to the total number of small cubes. If they do not, one of them is wrong.

For a dice, take the face you are asked about and write down every number that shares an edge with it in any of the views. In a view, the three faces shown all touch one another.

Cross off those numbers. The single number left among one to six is on the opposite face.

If too few neighbours are visible to leave one number, look again at the views: the same three numbers can be arranged round the corner in two ways, and only one of them is the view that is drawn.

Mistakes that cost marks

Using (l-2)(b-2)(h-2) for the cubes with one painted face. That expression counts the cubes with no paint at all.
Answering with the count of edge cubes when the question asks about face cubes, or the other way round. The two are equal for some blocks, which is what makes the mix-up easy to miss.
Forgetting that a cube on an edge between two corners has two painted faces, not one - it is on the outside in two directions.
Working out the unpainted cubes as the total minus the painted ones without checking what has been counted.
On a dice, offering a number that touches the given face. Those faces meet along an edge, so they cannot be opposite.
Ignoring which way round the view is drawn. A die and its mirror image give the same three numbers at a corner but not the same opposite faces.

Worked example

A block measuring 4 cm by 3 cm by 3 cm is painted all over and cut into cubes of side 1 cm. How many of the cubes have exactly two painted faces?

The block is 4 by 3 by 3, so it cuts into 4 x 3 x 3 = 36 cubes.
Take off a layer one cube thick from every side, leaving a block of 2 by 1 by 1 - so 2 of the cubes are right inside and have no paint at all.
The 8 corners have three painted faces each.
The cubes with two painted faces are the ones on an edge but not at a corner. Along the four edges of length 4 there are 2 each, giving 8, and along the eight edges of length 3 there is 1 each, giving 8. So there are 16.
Check: 8 corners with three faces, 16 with two, 10 with one, and 2 with none. That adds to 36, which is the number of cubes, so nothing has been missed.
The answer is 16.

Answer: 16

Practice questions with answers

A few Cubes and Dice questions with the full solution shown, so you can see how the method is applied before you attempt the timed set.

Question 1

A cube of side 4 is painted and cut into small cubes of equal size. Of these small cubes, how many have exactly one painted face?

A 12
B 40
C 8
D 24

Answer: Option D — with explanation

The block is 4 by 4 by 4 small cubes, so 64 small cubes in all. The 8 corners have three painted faces; a cube on an edge between two corners has two; a cube in the middle of a face has one; and the cubes with no face on the outside have none. Counting each kind in turn gives 8 with three painted faces, 24 with two, 24 with one, and 8 with none. Common mistakes - 12 is not the number of cubes of that kind: 12 - 40 is not the number of cubes of that kind: 40 - 8 is not the number of cubes of that kind: 8

Question 2

A block measuring 6 cm by 3 cm by 4 cm is painted all over. It is cut into cubes of side 1 cm. How many of the cubes have exactly one painted face?

A 12
B 28
C 8
D 44

Answer: Option B — with explanation

The block is 6 by 3 by 4 small cubes, so 72 small cubes in all. The 8 corners have three painted faces; a cube on an edge between two corners has two; a cube in the middle of a face has one; and the cubes with no face on the outside have none. Counting each kind in turn gives 8 with three painted faces, 28 with two, 28 with one, and 8 with none. Common mistakes - 44 is not the number of cubes of that kind: 44 - 8 is not the number of cubes of that kind: 8 - 12 is not the number of cubes of that kind: 12

Question 3

The same dice is shown in three positions below. Which number is on the face opposite the face showing 1?

Answer: Option C — with explanation

Two faces of a dice that meet along an edge cannot be opposite each other, so the way to settle this is to find which numbers share an edge with 1. Reading the three views, the numbers touching 1 are 2, 4, 5, 6, and none of them can be opposite it. The only number left is 3, so that is the face opposite. Common mistakes - 5 touches the face showing 1, so it cannot be opposite it: 5 - 4 touches the face showing 1, so it cannot be opposite it: 4 - 2 touches the face showing 1, so it cannot be opposite it: 2

Question 4

All the faces of a cube of side 6 are painted. The cube is then cut into 216 small cubes of equal size. How many of those cubes have exactly one painted face?

A 8
B 96
C 64
D 48

Answer: Option B — with explanation

The block is 6 by 6 by 6 small cubes, so 216 small cubes in all. The 8 corners have three painted faces; a cube on an edge between two corners has two; a cube in the middle of a face has one; and the cubes with no face on the outside have none. Counting each kind in turn gives 8 with three painted faces, 48 with two, 96 with one, and 64 with none. Common mistakes - 64 is not the number of cubes of that kind: 64 - 8 is not the number of cubes of that kind: 8 - 48 is not the number of cubes of that kind: 48

Question 5

A cuboid measuring 7 cm by 3 cm by 6 cm is painted on all its faces and cut into small cubes of side 1 cm. How many of those cubes have exactly 2 painted faces?

A 40
B 58
C 8
D 20

Answer: Option A — with explanation

The block is 7 by 3 by 6 small cubes, so 126 small cubes in all. The 8 corners have three painted faces; a cube on an edge between two corners has two; a cube in the middle of a face has one; and the cubes with no face on the outside have none. Counting each kind in turn gives 8 with three painted faces, 40 with two, 58 with one, and 20 with none. Common mistakes - 58 is not the number of cubes of that kind: 58 - 8 is not the number of cubes of that kind: 8 - 20 is not the number of cubes of that kind: 20

Frequently asked questions

How do I remember which count is which?

Count the directions a cube is on the outside in. A corner is on the outside in all three, an edge cube in two, a face cube in one, and a cube inside in none. Everything in these questions follows from that.

Is the number of corner cubes always 8?

Yes, for any block whose sides are all at least 2. A block has eight corners whatever its size, and each corner cube has three painted faces.

Why can I not just say a number that is next to the given face on the dice?

Because faces that meet along an edge are neighbours, and neighbours are never opposite. The opposite face is the one that touches none of the faces you can see alongside it.

The three views seem to fit two different dice. What then?

One of the two is the mirror image of the other. Read the views as they are drawn - which face is on the right for a given top and front - and only one of the two arrangements fits.