Question 1

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?

Accepted Answer

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.

Question 2

Three different positions of a dice are shown below. How many dots lie opposite 2 dots?

Accepted Answer

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.

Question 3

Three different positions X, Y and Z of a dice are shown in the figures given below. Which of the hidden numbers adjacent to 5 in position X is/are common to the hidden numbers adjacent to 5 in position Z?

Accepted Answer

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.

Question 4

Two positions of a dice are shown below. When there are two dots at the bottom, the number of dots at the top will be

Accepted Answer

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.

Question 5

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

Accepted Answer

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.

Question 6

Amongst the following figures, find the correct one, if it is known that the total number of dots on opposite faces of the cube shown is always 7.

Accepted Answer

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.

Question 7

A dice is numbered from 1 to 6 in different ways. If 1 is adjacent to 2, 4 and 6, then which of the following statements is necessarily true?

Accepted Answer

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).

Question 8

Two positions of a dice with 1 to 6 dots on its sides are shown below. If the dice is resting on the side with three dots, what will be number of dots on the side at the top?

Accepted Answer

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.

Question 9

Three different positions X, Y and Z of a dice are shown in the figures given below. Which number lies opposite 6?

Accepted Answer

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.

Question 10

Which number is on the face opposite 4, if the four different positions of a dice are as shown in the figure given below.

Accepted Answer

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

Question 11

Two positions of a dice are shown below. Identify the number at the bottom when the top is '3'?

Accepted Answer

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.

Question 12

Two positions of a block are given below. When 1 is at the top, which number will be at the bottom?

Accepted Answer

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.

Question 13

A cube has six different symbols drawn over its six faces. The symbols are dot, circle, triangle, square, cross and arrow. Three different positions of the cube are shown in figures X, Y, and Z. Which symbol is opposite the dot?

Accepted Answer

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. As analysed above, the cross lies opposite the dot.

Question 14

If the total number of dots on opposite faces of a cubical block is always 7, find the figure which is correct.

Accepted Answer

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.

Question 15

Four usual dice are thrown on the ground. The total of numbers on the top faces of these four dice is 13 as the top faces showed 4, 3, 1 and 5 respectively. What is the total of the faces touching the ground?

Accepted Answer

In a usual dice, the sum of the numbers on any two opposite faces is always 7. Thus, 1 is opposite 6, 2 is opposite 5 and 3 is opposite 4. Consequently, when 4, 3, 1 and 5 are the numbers on the top faces, then 3, 4, 6 and 2 respectively are the numbers on the face touching the ground. The total of these numbers = 3 + 4 + 6 + 2 = 15.

Question 16

The four different positions of dice are given below: Which number is on the face opposite 6?

Accepted Answer

From figures (i), (ii) and (iii), we conclude that the numbers 2, 3, 4 and 5 appear adjacent to 6. Clearly, the remaining number i.e. 1 will appear on the face opposite 6.

Question 17

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?

Accepted Answer

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.

Question 18

What is the opposite 3, if four different positions of dice are as shown below?

Accepted Answer

From figures (i), (iii) and (iv), we conclude that 2, 6, 1. and 5 appear adjacent to 3. Clearly, 4 will appear opposite 3.

Question 19

Two positions of a dice are shown. When 4 is at the bottom, what number will be on the top?

Accepted Answer

From figures (i) and (ii), we conclude that 2, 3, 5 and 6 he adjacent to 1. Therefore, 4 lies opposite 1. Hence, when 4 is at the bottom, then1 must be on the top.

Question 20

Two positions of a block are shown below. When 2 is at the bottom, which number will be at the top?

Accepted Answer

Number 3 is common to the two positions of the block. We assume the block in fig. (ii) to be rotated so that 3 appears at the same position as in fig. (i) i.e. on RHS face (i.e. on face II as per activity 1) and the numbers 5 and 2 move to the faces hidden behind the numbers 4 and 6 respectively [in fig. (i)]. Thus, the combined figure will have 3 on RHS face (i.e. face II), 4 on the Front face (i.e. face I), 6 on the Top face (i.e. face V), 5 on the Rear face (i.e. face III) and 2 on the Bottom face (i.e. face VI). Clearly, when 2 is at the bottom; then 6 is at the top.

Cubes and Dice Questions