Three positions of a dice are given. Based on them find out which number is found opposite the number 2 in the given cube.

Number 1 is common to both the positions of the dice. We assume the dice in fig. (ii) to be rotated so that 1 dot moves to the top face (i.e. face V as per activity 1) i.e. to the same position as in fig. (i) and 2 and 4 dots move to the faces hidden behind the faces with 3 and 5 dots respectively. Thus, the combined figure will have 1 dot on the Top (i.e. on face V), 5 dots on Front face (i.e. on face I), 3 dots on RHS face (i.e. face II), 4 dots on the Rear face (i.e. face III) and 2 dots on the LHS face (i.e. face IV). Clearly, 3 dots lie on the face opposite the face having 2 dots. Therefore, when there are 2 dots at the bottom, the number of dots at the top will be 3.

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.

Since 3 lies opposite 5 (as analysed above), it follows that 1, 4, 6 and 2 lie adjacent to 5. Out of these four numbers, the hidden numbers adjacent to 5 in position X are 6 and 2 and the hidden numbers adjacent to 5 in position Z are 1 and 4. Clearly, there is no number common.

From figures (ii) and (iii), we conclude that 1, 6, 3 and 4 dots lie adjacent to 5 dots. Therefore, 2 dots must lie opposite 5 dots. Conversely, 5 dots must lie opposite 2 dots.

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.

The number 2 is common to both the figures. We assume the parallelepiped in fig. (ii) to be rotated so that 2 appears at the same position as in fig. (i) i.e. on the RHS face and the numbers 6 and 3 move to the faces hidden behind the numbers 1 and 5 respectively [in fig. (i)]. Then, the combined figure will have 1 opposite 6 and 5 opposite 3. Thus, when 3 will be on the top, then 5 will appear at the bottom.

Since the total number of dots on opposite faces is always 7, therefore 1 dot must lie opposite 6 dots, 2 dots must lie opposite 5 dots and 3 dots must lie opposite 4 dots. In each of the two figures (2) and (4), 2 dots appear adjacent to 5 dots, and in fig. (3), 3 dots appear adjacent to 4 dots. Hence, these figures are incorrect. Therefore/only fig. (1) is correct.

If 1 is adjacent to 2, 4 and 6 then either 3 or 5 lies opposite to 1. So, the numbers 3 and 5 cannot lie opposite to each other. Hence, 3 is adjacent to 5 (necessarily).

From figures (i) and (ii) we conclude that 2, 6 and 4 dots appear adjacent to 3 dots. Hence, either 1 or 5 dots may appear opposite 3 dots. Thus, if the dice is resting on the side with three dots, then the number of dots on the side at the top is either 1 or 5.

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, 1 lies opposite 6.

From figures (i) and (iv) we conclude that 6, 5, 2 and 3 lie adjacent to 4. It follows that 1 lies opposite 4.