Cubes & Dice
Cubes and Dice is a common non-verbal reasoning topic of the ADRE 3.0 Grade IV paper. The topic actually contains two related ideas. In the cube questions a large cube is painted on all its faces and then cut into many smaller cubes, and the candidate must count how many of the small cubes have one, two, three or no painted faces. In the dice questions two or three views of a dice are shown and the candidate must find the number on the face opposite to a given face, or decide which faces can appear together. Both parts are fully logical and need no calculation beyond simple addition, but they demand a clear picture of a solid object in three dimensions. Once the standard rules are learned, these questions become quick and reliable sources of marks.
Meaning of Cube and Dice Questions
In a cube question a solid cube, whose edge is divided into n equal parts, is painted on the outside and then cut along these divisions into n cubed small cubes. The small cubes that lie at the corners keep three painted faces, those on the edges but not at the corners keep two painted faces, those on the surface but not on an edge keep one painted face, and those completely inside keep no painted face. In a dice question a standard dice is a cube whose six faces are marked with numbers or dots, and the opposite faces of a standard dice always add up to seven.
Important Rules and Formulas
- Total number of small cubes in an n by n by n cube is n cubed.
- Cubes with three painted faces are always the eight corners.
- Cubes with two painted faces are twelve times n minus two.
- Cubes with one painted face are six times the square of n minus two.
- Cubes with no painted face are the cube of n minus two.
- The sum of all the four counts is n cubed.
- In a standard dice, opposite faces add up to seven.
- 1 is opposite 6, 2 is opposite 5 and 3 is opposite 4 in a standard dice.
- A dice has eight corners, and three faces meet at every corner.
- Two faces that meet at an edge can never be opposite each other.
Types of Questions
The cube questions may ask for the number of cubes with exactly one painted face, exactly two painted faces, exactly three painted faces or no painted face, or they may ask for the total number of painted faces in the whole set. The dice questions show one, two or three positions of the same dice and ask for the number opposite a given face. Some dice questions give four numbers that appear around one corner and ask which number is opposite the fifth. A few questions combine both ideas by asking how a painted cube would look after being rolled.
How to Solve a Cube and Dice Question
- For a cube, note the value of n, the number of divisions on an edge.
- Apply the formulas for three, two, one and zero painted faces.
- Check that the four counts add up to n cubed.
- For a dice, remember that opposite faces add up to seven.
- Identify the common faces in two views and use them to find the opposite face.
Worked Examples
If a cube is cut into three parts along each edge, so that n is three, then the number of cubes with three painted faces is eight, with two painted faces is twelve, with one painted face is six, and with no painted face is one; the sum is twenty seven, which is three cubed. If a dice shows 1 on the top and 2 at the front, then 6 is at the bottom and 5 is at the back, because 1 and 6 are opposite and 2 and 5 are opposite. If two views of a dice share the number 3, then the three numbers around 3 in one view and the three numbers around 3 in the other view together account for all the remaining faces except the one opposite 3, which is easily found.
How This Topic Appears in the Examination
Cubes and dice is usually asked as one or two objective questions carrying one mark each. The cube sums are quick for a candidate who knows the formulas, while the dice questions need a careful comparison of the given views. The figures are small and can be redrawn on rough paper if needed. A candidate who has learned the standard dice rule and the four cube formulas can answer these questions in a very short time and use the remaining time for the arithmetic part of the paper.
Memory Tips
Remember the four cube formulas as a set of corner, edge, face and inner cubes, since this order makes them easy to recall. Always check your answer by adding the four counts and seeing whether they give n cubed. For a dice, memorise the opposite pairs 1 and 6, 2 and 5, 3 and 4. When two views are given, look for the face that appears in both, because the faces touching it in the two views help to fix the whole dice. Draw a small cube and mark the painted faces while solving, if that makes the picture clearer.
Common Mistakes to Avoid
- Confusing the number of painted faces with the number of small cubes.
- Forgetting that the corner cubes have three painted faces.
- Using the wrong value of n in the formulas.
- Assuming that any two faces of a dice can be opposite.
- Reading the dots of a dice as numbers in the wrong order.
How to Prepare This Topic
- Learn the four cube formulas with the help of a diagram.
- Learn the standard opposite pairs of a dice by heart.
- Practise at least fifteen cube and dice questions every day.
- Draw the cube or dice on rough paper before answering.
- Time yourself so that each question takes less than a minute.
Additional Solved Examples
If a cube is cut into four parts along each edge, so that n is four, then the corner cubes are eight, the edge cubes are twenty four, the one-face cubes are twenty four, and the inner cubes are eight; the sum is sixty four, which is four cubed. If a dice shows 2 on the top and 3 at the front, then 5 is at the bottom and 4 is at the back. If a dice is rolled and the number on the top changes from 1 to 3, then the number at the bottom changes from 6 to 4, because the opposite pairs remain fixed however the dice is turned. If a cube is painted only on two opposite faces and then cut, the painted small cubes lie only on those two faces.
Advanced Tips
When a cube is painted on all faces but cut only along one direction, the number of painted faces is different from the fully cut case, so always read the question carefully before applying the formulas. In dice questions, if two views have no face in common, find a third view or use the rule that the three numbers meeting at a corner can be arranged only in two ways. Remember that a standard dice may be shown in any orientation, but the opposite pairs never change. If a question gives the sum of the dots on a face, recall that the total of all six faces of a standard dice is twenty one. Finally, when counting cubes with exactly two painted faces, do not forget that edge cubes at the corners have three painted faces and must be excluded.
Related Non-Verbal Topics
Cubes and dice is closely related to Counting of Figures, Embedded Figures and Pattern Completion. Counting of figures also deals with counting shapes in a drawing, and the cube question is a three-dimensional version of the same skill. Embedded figures asks you to find a hidden shape, which helps in imagining the small cubes inside the large one. Pattern completion asks for the missing part of a design, and a dice with a hidden face is a similar problem in three dimensions. Practising these topics together strengthens spatial imagination, which is the main demand of the non-verbal section.
Quick Revision Checklist
- Total small cubes in an n by n by n cube is n cubed.
- Three-face cubes are eight, two-face cubes are twelve times n minus two.
- One-face cubes are six with n minus two squared, zero-face cubes are n minus two cubed.
- Standard dice opposite pairs are 1-6, 2-5 and 3-4.
- Faces meeting at an edge can never be opposite.
- Check the cube counts by adding them to get n cubed.
Frequently Asked Questions
What is a cube question?
A large cube is painted on all faces and cut into small cubes, and you are asked to count the small cubes with one, two, three or no painted faces.
How many small cubes have three painted faces?
Whatever the value of n, the cubes with three painted faces are always the eight corner cubes of the large cube.
What is the standard rule for a dice?
In a standard dice the numbers on opposite faces add up to seven, so 1 is opposite 6, 2 is opposite 5 and 3 is opposite 4.
Can two faces meeting at an edge be opposite?
No. Two faces that meet at an edge of the cube can never be opposite, because opposite faces never touch each other.
How do I check my cube answer?
Add the numbers of cubes with three, two, one and no painted faces; the total must be equal to n cubed, which is the total number of small cubes.