Dice

A Dice is a cube with 6 faces each of which contains a digit or a pattern denoting a digit from 1 to 6. It is, however, not uncommon for the faces of a dice to have different kinds of symbols. These puzzles based on dice are quite common in aptitude sections of various tests including GATE. Therefore, it is important to understand how to approach these kinds of problems.
Here is a sample representation of a dice :

The Dice shown above has 6 faces which are as follows:

1. EFHG
2. ABCD
3. ABFE
4. DCHG
5. EADG
6. GDCH

Constructed Dice
A Constructed Dice is one in which we are provided with front facing representation(s) of the dice from which we have to infer the information required to answer the question.
Usually we may be given two or more front facing figures of such a dice which will show us 3 of the 6 faces and using the information about the faces obtained from these figures we have to construct all or part of the dice. This constructed dice is then used to answer the question(s) asked.

Approach
To approach these problems we can make a mental dice or a diagram which we can use to get all the positions on the dice. Along with these there are a few useful observations that I have listed here:

1. If we are given 2 figures of a dice that have a common face in the same position then we will have 2 cases:
   Case 1: If we have all faces that are adjacent to the single common face then we will have all the 4 faces adjacent to the single common face. The faces in the first figure will be opposite to the faces in the second figure in the same position.
   Case 2: If we have one adjacent face common and the third face different in the second figure then we get 4 of the 3 adjacent faces to the common face then the face not in the 2 figures will be opposite to the common adjacent face.
2. If we are given 2 figures of a dice that have a common face but in different positions then we can perform an imaginary rotation on it to get it in the same position as first figure and then we proceed as in the previous point.
   In this manner for multiple configurations we can select any two of them focus on a common face and based on the above 2 points correctly find its adjacent faces and mark them on a diagram.
   Using this procedure we can find the required information about the dice and then answer the question.

Example 1
There are two figures of dice given below. Using the information from the figures find out which number will be facing opposite to two.

Solution:
We have 1 in a fixed position and the 2 adjacent faces are different in both the figures. Therefore, from Case 1 of Point 1 above we have 2 opposite to 4 and 6 opposite to 3 and 2,6,3 and 4 are adjacent to 1.
Thus, 4 is the required answer.

Example 2
Which symbol will be opposite to the symbol @.

Solution:
Here we will make use of figures 2 and 3 where # is the common symbol in both at the same positions with no other common faces. Thus, the two adjacent faces in second figure will be opposite to the symbols in the third figure that are at the same position.
From this we can conclude that @ is opposite to - and $ is opposite to *.
Thus, - is the required answer.

Deconstructed Dice
A deconstructed dice is one in which we are provided with a flattened out version of a dice which may have all or some of the symbols and to answer the question we have to visualize the constructed version of the dice.

The question here will usually focus on telling about the face that is adjacent or opposite to a particular face. As the representation is flattened out the adjacent and opposite faces appear disconnected and away from the face considered. Therefore, it is difficult to answer the question directly based on the flattened out representation and we need to construct the dice to answer the question.

Approach
A question which gives us a flattened representation may ask us one of the following 3 things corresponding to a particular face:

1. Opposite Face
2. Adjacent Face
3. A constructed version of the die

To find the opposite face we can modify the given flattened representation to a form from which it will be more easy to find the opposite face which will be the face that is 1 face apart in the same row or column.
The adjacent faces can be found using the constructed version of the dice with the constructed version containing the face in question.

Example 1
Given a flattened representation of a dice if construct a standard dice out of it which number will be opposite to the face containing 3.

Solution:
We change the given flattened representation to the following form:

From this representation we can see that 1 and 3 are separated by the face containing the number 6. Therefore, 1 is the face opposite to 3.
Thus, answer is 1.

Example 2:
Given a flattened representation of a dice if construct a standard dice out all the faces that are adjacent to the face containing the number 1.

Solution:
We will construct a dice and draw its 2 front facing representations in which 1 will be the top face:

From this we can clearly deduce the four faces adjacent to 1 which are: 4, 5, 6, 2 which is the required answer.