Cubes and Dice for the Assam Police Constable (AB & UB) Exam
Cubes and Dice is a non-verbal reasoning topic of the SLPRB Assam Police Constable (AB & UB) examination. It has two closely related parts. In the dice part two or three views of the same dice are shown and the candidate must find the number or figure on the face opposite to a given face. In the cube part a large cube painted on all sides is cut into a number of smaller cubes and the candidate must count how many small cubes have one painted face, two painted faces, three painted faces or no painted face at all. The topic needs no calculation beyond simple counting, but it needs clear visualisation. Once the rules are learnt, the questions are solved in a few seconds. This page explains the rules of opposite faces, the standard cube formulas, worked examples, shortcuts and the common traps. The material follows the standard reasoning reference books used for Assam recruitment examinations and is reviewed before it is published.
What the Topic Means
A dice is a cube whose faces carry dots or numbers, and in the given views the same dice is turned into different positions. Because the hidden faces cannot be seen, the only way to answer is to compare the views and decide which face lies opposite which. A cube question is a counting question in which a big cube is painted and then cut, and the candidate must count the small cubes according to how many of their faces are painted. Both parts rest on one fact that should be remembered: a cube has six faces, every face has exactly one opposite face, a corner cube always has three painted faces, and a cube on an edge has two.
Rules for Finding the Opposite Face of a Dice
- When three faces are visible and two of them are common in two views, the face that has not appeared is opposite to the face that changed.
- If two views have one common face, the two faces adjacent to the common face in the first view are opposite the two adjacent faces in the second view.
- In any view the four side faces are in a ring, so a face is opposite the one that is two steps away in the ring.
- Touching faces are never opposite; a face and the one opposite to it can never be seen together.
- For a die whose opposite faces add to seven, the sum of any two opposite faces is seven.
Rules for a Painted Cube Cut into Small Cubes
- A cube painted on all sides and cut into n by n by n small cubes has three painted faces only on its eight corners, so the number with three painted faces is always eight.
- The cubes with two painted faces lie on the twelve edges and number twelve multiplied by n minus two.
- The cubes with one painted face lie on the six faces and number six multiplied by n minus two squared.
- The cubes with no painted face lie inside and number n minus two cubed.
- The total of these four numbers is always n cubed, so the count can always be checked.
A Step by Step Method
- For a dice question, first find the common face in the two views and note the faces next to it.
- List the opposite pairs one by one and write them down before looking at the options.
- For a cube question, first find n, the number of small cubes along one edge.
- Apply the four formulas for three, two, one and no painted faces.
- Add the four counts and check that the total equals n cubed.
Worked Examples
Example one: in two views of a dice the faces 1, 2 and 3 and then 1, 4 and 5 are seen. Since 1 is common, the faces 2 and 3 are opposite 4 and 5 in the same order, so 2 is opposite 5 and 3 is opposite 4. Example two: if 1, 2 and 3 are seen together then none of them is opposite another, so 1, 2 and 3 are mutually adjacent. Example three: a cube is painted and cut into 27 small cubes; the number 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. Example four: a cube cut into 64 small cubes has eight cubes with three painted faces, twenty-four with two painted faces, twenty-four with one painted face and eight with none. Example five: if the sum of opposite faces is seven and 2 is seen, then the opposite face is 5, and the remaining faces are found from the other view.
How This Topic Appears in the Examination
The topic usually appears as two to four objective questions of one mark each. The common forms are to find the number opposite a given face, to find which face is at the bottom or on one side, to count the small cubes with a stated number of painted faces, and to count the number of small cubes left after a cut. The options are small numbers, so one wrong formula flips the answer.
Shortcuts and Speed Methods
- Memorise the four painted-cube formulas and the sum-to-seven rule.
- Write the opposite pairs in a line before reading the options.
- For a painted cube, n is always the edge count and the total must equal n cubed.
- Remember that corner cubes have three painted faces and edge cubes have two, whatever the value of n.
- For a dice, look for the common face first; it gives the opposite pairs quickly.
Common Mistakes to Avoid
- Treating two faces that are seen together as opposite; they are always adjacent.
- Using the wrong value of n in the cube formulas.
- Adding the four counts and not checking that the total equals n cubed.
- Forgetting that a standard die always sums to seven on opposite faces.
- Assuming the dice faces are numbered in the same order in every view.
Preparation Plan
- Learn the four painted-cube formulas by heart.
- Learn the method of finding opposite faces from two views.
- Practise a dice question and a cube question every day.
- Draw the opposite pairs and the cube counts in a table.
- Revise the formulas once a week until they are recalled instantly.
More Practice Questions
- Two views of a dice show 1, 2, 3 and then 2, 3, 4; find the face opposite 1.
- A cube is painted and cut into 125 small cubes; count the cubes with two painted faces.
- A cube is cut into 216 small cubes; find the number with no painted face.
- Find the opposite of a given face on a standard die that sums to seven.
- Count the small cubes with exactly one painted face in a four by four by four cut.
Quick Revision Checklist
- Find the common face in the two views.
- Write the opposite pairs in a line.
- Find n from the edge of the big cube.
- Apply the four painted-face formulas.
- Check that the four counts add up to n cubed.
Cubes, Dice and Spatial Thinking
Cubes and dice test the ability to hold a three-dimensional object in mind and to turn it without touching it. The same spatial thinking helps in paper folding and in embedded figures, where a shape has to be imagined from a flat drawing. The formulas of the painted cube are few, and once they are fixed in memory the questions become mechanical. In the dice part the trick is to compare the views patiently and to write down the pairs rather than to guess from the picture.
Frequently Asked Questions
What is the main rule of a standard die?
On a standard die the numbers on opposite faces always add up to seven, so if one face is known the opposite face is found by subtracting it from seven.
How do I find the opposite face from two views?
Find the face that is common to both views; the faces touching it in the first view are opposite the faces touching it in the second view, and the remaining unseen face is opposite the changed face.
How many small cubes have three painted faces?
Always eight, because a cube has eight corners, whatever the number of small cubes along one edge.
How do I count the cubes with two painted faces?
The cubes on the twelve edges of the big cube have two painted faces, and their number is twelve multiplied by n minus two.
Is cubes and dice important for Assam Police?
Yes. The rules are fixed and the counting is short, so it is a reliable source of marks in the reasoning paper.