Cubes and Dice tricks and concepts

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Basic Concepts:

What is a Dice?
A Dice is a cube. In a cube, there are six faces. Let us study some basic facts about Cube using the given figure.
The six faces in the cube are– ABCG, GCDE, DEFH, BCDH, AGEF and ABHF.

Four faces are adjacent to one face
There are pairs of opposing faces e.g. Opposite of DEFH is ABCG and so on
CDEG is the upper face of the cube
ABHF is the bottom face of the cube

Important Facts:

A cube has 6 square faces or sides
A cube has 8 points (vertices)
A cube has 12 edges
Only 3 sides of a cube are visible at a time (known as “Joint Sides”) and these sides can never be on the opposite side of each other
Things that are shaped like a cube are often referred to as ‘cubic’
Most dice are cube shaped, with the numbers 1 to 6 on the different faces.

Certain Basic Rules:

There are certain dice rules in reasoning which can be sued to solve dice-based questions:

Rule No. 1:
Two opposite faces of the dice cannot be adjacent to each other.
E.g. Two positions of a dice are shown below.

Here, faces with number 4, 3, 6 and 1 are adjacent to the face number 2.
Therefore, the 4,3,6,1 can’t be opposite to the face number 2.
Therefore, face number 5 is opposite to the face number 2.

Rule No. 2:
If two dice are shown as below, and one of the two common faces (Face number 4) is in the same position, then the remaining faces will be opposite to each other.
E.g.: Two dice are shown below

In both the diagrams, two faces numbered 1 & 3 are common.
Also, 5 & 6 are remaining faces. Hence, face which is number 5 is opposite to the face number 6.

Rule No. 3:
If in 2 different positions of the dice, the positions (different), the position of the face that’s common is the same, and then the opposite faces of the faces that remain will be in the same positions.
E.g.:

In both the positions, face number 1 is common for both dice is same.
Therefore, the opposite of 4 is 2 and the opposite of 5 is 6.

Rule No. 4:
If 2 positions of a die are given (Different) and it is also stated that common face is different then the face opposite to the given common face would be that which is not shown on any given face in the 2 given positions. It is also to be noted that the opposite face of the faces that are left cannot be the same
E.g.:

Note, in the above shown dice, the face having value 6 is not in the similar position.
The face numbered 1 is not shown.
So, the face opposite to the face with number 6 is 1.
Also, the opposite face of 3 is the face with number 2 and the opposite to face numbered 5 is the face with number 1.

Deconstructing a cube:

When we are given a dice it is somewhat difficult to visualize it in 3-D so what we do is that we flatten the cube. We can form a cube that has been flattened where we can visualize, the square at the farthest end will give the top of the cube and the square that is the middle will form the base of the given dice. The given figures below can help you understand the theory stated above.

The rest of the square will give the adjacent sides of the dice. Note that we have to clearly visualize the adjacent sides and we have to figure out what exactly the question is asking. The flattening of dice is the easiest way that we can use to solve the dice problems.

Now that you know the basic concepts on cubes and dice reasoning tricks. Let us study a few solved examples on dice reasoning Click here & solve some questions on dice.

Some Examples and solutions

Example 1: When the given figure is folded to form a cube then which face is opposite to the face with 2?

A. 2
B. 4
C. 1
D. 6

Solution : Now, this is a peculiar type of flattened cube. However, the rules to solve the questions remain the same. Now, if you want to find the face that is opposite to the face with 2, you should consider it as the base. So, the top face (opposite face) will be the 1.

Example 2: Choose a box that is similar to the box formed from the given sheet of paper.

A. A & C only
B. A & D only
C. B & C only
D. C & D only
solution: Looking at the figure given, the two square sides will form the top and bottom faces of the cube. Also, the two rectangular blank strips will form the opposite faces to each other. In figure B and figure D, the two filled rectangular faces have been shown adjacent to each other, so these figures are obviously not accurate. However, options A & C are completely possible.

Example 3: Choose a box that is not different to the box formed from the given sheet of paper (X).

A. 1 and 2 only
B. 2 only
C. 1, 2 and 3 only
D. 1, 2, 3 and 4 only
solution: When the given sheet in the fig. (X) is folded to form the hollow cube, then letter '6' appears opposite letter '2', letter '5' appears to be opposite letter '3' and letter '1' appears opposite letter '4'. Hence, the hollow cube in the figure (1) that shows letter '2' adjacent to letter '6', the hollow cube in figure (3) which depicts letter '3' adjacent to number '5' and the hollow cube in the figure (4) which depicts letter '4' adjacent to letter '1' is not at all possible. Therefore, only the hollow cube in the figure (2) is possible.

Example 4: Choose a box that is similar to the box formed from the given sheet of paper (X).

A. 2 only
B. 3 only
C. 1 only
D. 4 only
Solution: When the given sheet in the figure (X) is folded to form a hollow cube, the K face will appear opposite to the face that features a triangle, the face that has ? will appear opposite to one of the two blank faces & to the face that bears a '÷' sign appears opposite to the other blank face given in the figure. Obviously, the cubes shown in figure (1) & figure (4) are not at all possible because they have the K face adjacent to the face that features the triangle. Although the hollow cube shown in the figure (3) has faces that will appear adjacent to each other but the hollow cube formed by folding the sheet in the given figure(X) cannot be rotated to form the fig. (3). Therefore, the hollow cube in figure (3) cannot be formed. Hence, only the hollow cube shown in the figure (2) is possible.

Example 5: Choose a box that is similar to the box formed from the given sheet of paper (X).

A. 3 only
B. 2 only
C. 1 only
D. 4 only
Solution: When the given sheet depicted in the figure (X) is folded to form a hollow cube, the two half-shaded faces will lie opposite to each other, the face that features a square will lie opposite to one of the two blank faces in the sheet given, and the face with a circle will lie opposite to the other blank face in the sheet. So, the hollow cubes shown in the figure (2) and figure (3) that have two half-shaded faces that are adjacent to each other, is not possible to be made by folding the sheet depicted in the figure (X). Although the hollow cube depicted in the figure (4) has faces that may appear to be adjacent to each other but the hollow cube formed by folding the given sheet in the figure (X) cannot be rotated to form the hollow cube in the figure(4). Therefore, only the hollow cube in fig.ure(1) is possible.