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?

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.

From figures X and Y, we conclude that dot, circle, square and cross lie adjacent to the triangle. Therefore, the arrow must lie opposite the triangle. From figures X and Z, we conclude that dot, triangle, arrow and cross lie adjacent to the circle. Therefore, the square must lie opposite the circle. Thus, the arrow lies opposite the triangle, the square lies opposite the circle and consequently, the cross lies opposite the dot.

Since the square lies at the top of fig. (Y) and the circle lies opposite the square (as analysed above), so the circle occurs at the bottom of fig. (Y).

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

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.

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.

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 figures (i) and (ii), we conclude that the numbers 1, 4, 3 and 5 lie adjacent to the number 6. Clearly, the number 2 lies opposite 6 and conversely 6 lies opposite 2.

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.