1. What will be the number at the bottom, if 5 is at the top; the two positions of the dice being as given below:

A. 1 B. 2 C. 3 D. 6

Answer - Option B
Explanation - From figures (i) and (ii), it is clear that 4, 1, 3 and 6 he adjacent to 2. Therefore, 5 must lie opposite 2. Thus, if 5 is at the top, then 2 must be at the bottom.
2. If the total number of dots on opposite faces of a cubical block is always 7, find the figure which is correct.

A. 1 B. 2 C. 3 D. 4

Answer - Option B
Explanation - Since the total number of dots on opposite faces is always 7, therefore, 1 dot appears opposite 6 dots, 2 dots appear opposite 5 dots and 3 dots appear opposite 4 dots.
3. Two positions of a block are given below. When 1 is at the top, which number will be at the bottom?

A. 2 B. 3 C. 4 D. 6

Answer - Option D
Explanation - Number 2 is common to the two positions of the dice. We assume the dice in fig. (ii) to be rotated so that 2 remains on the top face (i.e. face V as per activity 1) and the numbers 3 and 6 move to the faces hidden behind 5 and 1 respectively [in fig. (i)]. Then, clearly, 5 lies opposite 3 and 6 lies opposite 1. Hence, when 1 is at the top, then 6 will be at the bottom.
4. What number is opposite 3 in the figure shown below? The given two positions are of the same dice whose each surface bears a number among 1, 2, 3, 4, 5 and 6.

A. 2 B. 4 C. 5 D. 6

Answer - Option C
Explanation - From figures (i) and (ii), we conclude that 2, 6, 1 and 4 appear adjacent to 5. Therefore, 3 must appear opposite 5, Conversely, the number opposite 3 is 5.
5. Two positions of a dice are shown below. Identify the number at the bottom when the top is '3'?

A. 2 B. 4 C. 5 D. 6

Answer - Option C
Explanation - The number 6 is common to both the positions of the dice. We assume the dice in fig. (ii) to be rotated so that 6 remains on the FR-LH face (i. e. on face IV as per activity 1) and the numbers 1 and 3 move to the faces hidden behind 4 and 5 respectively [in fig. (i)]. Then, 5 lies on FR-RH face (i.e. face I), 4 lies on Top face {i.e. face V), 3 lies on RR-LH face (i.e. face III) and 1 lies on Bottom face (i.e. face VI). Thus, 3 lies opposite 5. Hence, when the top is 3, then the number at the bottom is 5.

1. Three different positions X, Y and Z of a dice are shown in the figures given below. Which number lies at the bottom face in position X?

A. 2 B. 3 C. 6 D. Cannot be determined

Answer - Option B
Explanation - From positions X and Y we conclude that 1, 5, 6 and 3 lie adjacent to 4. Therefore, 2 must lie opposite 4. From positions Y and Z we conclude that 4, 3, 2 and 5 lie adjacent to 6. Therefore, 1 must lie opposite 6. Thus, 2 lies opposite 4, 1 lies opposite 6 and consequently 5 lies opposite 3.
As analysed above, the number on the face opposite 5 is 3. In position X, since 5 lies on the top, therefore 3 must lie at the bottom face.
2. A dice is numbered from 1 to 6 in different ways. If 2 is opposite to 3 and adjacent to 4 and 6, then which of the following statements is necessarily true?

A. 1 is opposite to 5 B. 4 is opposite to 6 C. 4 is adjacent to 2 and 6 D. 1 is adjacent to 2 and 3

Answer - Option D
Explanation - If 2 is opposite to 3, then 1 cannot lie opposite to either of the two numbers - 2 or 3. Hence, 1 is necessarily adjacent to both 2 and 3.
3. Two positions of a dice are shown below. When number 1 is on the top, what number will be at the bottom?

A. 2 B. 3 C. 5 D. Cannot be determined

Answer - Option C
Explanation - Number 6 is common to both the positions of the dice. We assume the dice in fig. (ii) to be rotated so that 6 remains on the top face (i.e. face V as per activity 1) and the number 4 in fig. (ii) moves to the FR-RH face (i.e. face I) as in fig. (i), then 5 will move to the RR-RH face (i.e. face II). Clearly, 5 (which lies on face II) and 1 (which lies on face IV) will be opposite to each other. So, when 1 is on the top, then 5 will be at the bottom.
4. Two positions of a cube are shown below. When the number 4 will be at the bottom, then which number will be at the top?

A. 3 B. 5 C. 6 D. Cannot be determined

Answer - Option A
Explanation - Number 1 is common to both the positions of the dice. We assume the dice in fig. (ii) to be rotated so that 1 remains on the RHS face (i.e. face II as per activity 1) and the numbers 5 and 3 move to the faces hidden behind 2 and 4 respectively (in fig. (i). Then, clearly 3 lies opposite 4. Hence, when 4 is at the bottom then 3 mist lie on the top.
5. Two positions of a block are shown below: When six is at the bottom, what number will be at the top?

A. 1 B. 2 C. 4 D. 5

Answer - Option D
Explanation - From figures (i) and (ii) we conclude that the number 1, 2, 3 and 4 appear adjacent to 6. Thus, the number 5 will appear opposite 6. Therefore, when six is at the bottom, then 5 will be at the top.

Direction [1 - 5] : The sheet of paper shown in the figure (X) given on the left hand side, in each problem, is folded to form a box. Choose from amongst the alternatives (1), (2), (3) and (4), the boxes that are similar to the box that will be formed.
1. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 only B. 1 and 3 only C. 1, 3 and 4 only D. 1, 2, 3 and 4

Answer - Option C
Explanation - When the sheet in fig. (X) is folded, then one of the faces of the cube formed will be of the form and this face will lie opposite the face bearing a square. Also, one of the blank faces lies opposite another blank face and the third blank face lies opposite the face bearing an '=' sign. Clearly, all the three blank faces cannot appear adjacent to each other. So, the cube shown in fig. (2) which has all the three blank faces adjacent to each other cannot be formed. Hence, only the cubes shown in figures A, C and D can be formed.
2. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 only B. 2 only C. 1 and 3 only D. 1, 2, 3 and 4 only

Answer - Option B
Explanation - The fig. (X) is similar to the Form III. So, when the sheet in fig. (X) is folded to form a cube, then 'F' appears opposite 'B', 'E' appears opposite 'C' and 'A' appears opposite 'D' Therefore, the cube in fig. (1) which shows 'F' adjacent to 'B' the cube in fig. (3) which shows 'E' adjacent to 'C' and the cube in fig. (4) which shows 'A' adjacent to 'D' cannot be formed. Hence, only the cube in fig.(2) can be formed.
3. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 only B. 2 only C. 3 only D. 4 only

Answer - Option B
Explanation - The fig. (X) is similar to the Form III. So, when the sheet in fig. (X) is folded to form a cube, then the half-shaded face appears opposite to the face bearing a rhombus, the face with a black circle appears opposite to one of the two blank faces and the face with a '+' sign appears opposite to the other blank face. Clearly, the cubes shown in figures (1) and (4) cannot be formed since they have the half-shaded face adjacent to the face bearing the rhombus. Also, though the cube shown in fig. (3) has faces that can appear adjacent to each other but the cube formed by folding the sheet in fig. (X) cannot be rotated to form fig. (3). Hence, the cube in fig. (3) cannot be formed. Thus, only the cube shown in fig. (2) can be formed.
4. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 1 only B. 2 only C. 3 only D. 4 only

Answer - Option A
Explanation - The fig. (X) is similar to the Form II. So, when the sheet shown in fig. (X) is folded to form a cube then the two half-shaded faces lie opposite to each other, the face bearing a square lies opposite to one of the two blank faces and the face bearing a circle lies opposite to the other blank face. Therefore, the cubes shown in figures (2) and (3) which have the two half-shaded faces adjacent to each other, cannot be formed by folding the sheet shown in fig. (X). Also, though the cube shown in fig. (4) has faces that can appear adjacent to each other but the cube formed by folding the sheet in fig. (X) cannot be rotated to form the cube in fig. (4). Hence, only the cube in fig. (1) can be formed.
5. Choose the box that is similar to the box formed from the given sheet of paper (X).

A. 2 and 3 only B. 1, 3 and 4 only C. 2 and 4 only D. 1 and 4 only

Answer - Option B
Explanation - The fig. (X) is similar to the Form V. So, when the sheet in fig. (X) is folded to form a cube, then the face bearing a dot lies opposite to one of the shaded faces. Therefore, the cube shown in fig. (2) which has both the shaded faces adjacent to the face bearing the dot, cannot be formed. Hence, the cubes shown in figures (1), (2) and (4) can be formed.