A dice is rolled twice and the two positions are shown in the figure below. What is the number of dots at the bottom face when the dice is in position (i)?

From figures (i), (ii), (iv) we conclude that 5, 6, 1 and 2 lie adjacent to 4. Hence, 3 must lie opposite 4 and vice-versa. In fig. (iii), 3 is at the top and consequently 4 must lie at the bottom face.

Clearly, the six faces are labelled as

Face I -> a, Face IV -> b, Face III -> c, Face II -> d, Face V -> e, Face VI -> f

Therefore 'a' appears opposite 'c'.

Hence, when 'c' is at the top, then 'a' will be at the bottom.

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.

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.

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

From figures (i), (iii) and (iv), we conclude that 4, 5, 6 and 1 lie adjacent to 3. Therefore, 2 must lie opposite 3. Now, we have 1 opposite 5 and 2 opposite 3. Hence, 4 must lie opposite 6.

We shall assume the dice in fig. (ii) to be rotated so that the 5 dots appear at the same position as in fig. (i) i.e. on RHS face (i.e. on face II as per activity 1) and 1 dot appears at the same position as in fig; (i) i.e. on Front face (i.e. on face I). Then, from the, two figures, 2 dots appear on the top face (i.e. on face V) and 4 dots appear on the Bottom face (i.e. on face VI).

Since, these two faces are opposite to each other, therefore, two dots are contained on the face opposite to that containing four dots.

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.

If 1 is opposite to 5 and 2 is opposite to 3, then 4 definitely lies opposite to 6. Therefore, 2 cannot lie opposite to any of the two numbers - 4 or 6. Hence, 2 necessarily lies adjacent to both 4 and 6.

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

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.