Question 1

Two positions of a dice are shown below. When number 1 is on the top, what number will be at the bottom?

Accepted Answer

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.

Question 2

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 arrow?

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 symbol opposite the arrow is the triangle.

Question 3

If total number of cuts is 10 then find the minimum number of pieces that can be obtained.

Accepted Answer

If total number of cuts is 10 then minimum number of pieces is 11 when cut is made in one plane only.

Question 4

If total number of cuts is 10 then find the maximum number of pieces that can be obtained.

Accepted Answer

If total number of cut is 10 then for maximum number of pieces these cuts have to be well distributed in three planes. For 10 cuts, 3,3 and 4 is the distribution of cuts. Hence total number of pieces is (3 + 3)(3 + 1)(4 + 1) = 4 x 4 5 = 80

Question 5

If total number of cuts is 20 then find the ratio of maximum and minimum of pieces that can be obtained.

Accepted Answer

For maximum number of pieces cuts has to be 6, 7 and 7 and maximum number of pieces is (6 + 1)(7 + 1)(7 + 1) = 7 x 8 x 8 = 448. Minimum number of pieces is 20 + 1 = 21. Hence required ratio is 448:21

Question 6

If total number of pieces (Smaller cubes/cuboids) is 45 then find the possible number of cuts.

Accepted Answer

If 45 = 1 x 1 x 45 then we require only 44 cuts in one plane. If 1 x 3 x 15 then we require 2 cuts in one plane and 14 cuts in other plane so total number of cuts is 2 + 14 = 16. If 1 x 5 x 9 the we require 4 cuts in one plane and 8 cuts in other plane so total number of cuts is 4 + 8 = 12 If 3 x 3 x 5 then we require 2 cuts in one plane, 2 cuts in 2nd plane and 4 cuts in 3rd plane so total number of cuts is 2 + 2 + 4 = 8.

Question 7

Find the maximum number of cuts required to get 50 pieces.

Accepted Answer

For maximum number of cuts it has to be in one cut only, so number of cuts is 49

Question 8

Find the minimum number of cuts required to get 50 pieces.

Accepted Answer

For minimum number of cuts we will get 50 from 2 x 5 x 5 and cuts is 1 + 4 + 4 = 9

Question 9

How many of the cubes have 0 faces painted?

Accepted Answer

Since total number of cubes is hence in the formula we will substitute n = 6 Number of the cubes with 0 faces painted is (6 - 2)3 = 43 = 64

Question 10

How many of the cubes have 2 most faces painted?

Accepted Answer

Since total number of cubes is hence in the formula we will substitute n = 6 Number of the cubes with 2 faces painted is 6(6 - 2)2 = 6 x 16 = 96

Question 11

How many of the cubes have at most faces painted?

Accepted Answer

Since total number of cubes is hence in the formula we will substitute n = 6 At most 2 faces painted means number of cubes with 0 face painted + number of cubes with 1 face painted + number of cubes with 2 face painted = 64 + 48 + 96 = 208

Question 12

How many of the cubes have at least 2 faces painted?

Accepted Answer

Since total number of cubes is hence in the formula we will substitute n = 6 At least 2 faces painted means number of cubes with 2 face painted + number of cubes with 3 face painted = 96 + 8 = 104.

Question 13

How many of the cubes have 0 faces painted?

Accepted Answer

Out of 6 faces of 3 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 1 Number of vertices with 2 faces exposed (Painted) is 3 Number of vertices with 1 faces exposed (Painted) is 3 Number of vertices with 0 faces exposed (Painted) is 1 Number of sides with 2 sides exposed (Painted) is 3 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 3 From the above observation Number of cubes with 3 faces Painted is 1 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18 Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 1- 18 - 108 = 216 In other words number of cubes with 0 painted is (7 - 1)3 = 216. From the above explanation number of the cubes with 0 faces painted is 216.

Question 14

How many of the cubes have 2 faces painted?

Accepted Answer

Out of 6 faces of 3 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 1 Number of vertices with 2 faces exposed (Painted) is 3 Number of vertices with 1 faces exposed (Painted) is 3 Number of vertices with 0 faces exposed (Painted) is 1 Number of sides with 2 sides exposed (Painted) is 3 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 3 From the above observation Number of cubes with 3 faces Painted is 1 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18 Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 1- 18 - 108 = 216 In other words number of cubes with 0 painted is (7 - 1)3 = 216. From the above explanation number of the cubes with 2 face painted is 18.

Question 15

How many of the cubes have at most faces painted?

Accepted Answer

Out of 6 faces of 3 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 1 Number of vertices with 2 faces exposed (Painted) is 3 Number of vertices with 1 faces exposed (Painted) is 3 Number of vertices with 0 faces exposed (Painted) is 1 Number of sides with 2 sides exposed (Painted) is 3 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 3 From the above observation Number of cubes with 3 faces Painted is 1 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18 Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 1- 18 - 108 = 216 In other words number of cubes with 0 painted is (7 - 1)3 = 216. From the above explanation number of the cubes with at most 2 faces painted is 216 + 108 + 18 = 342. Or else 343 - 1 = 342

Question 16

How many of the cubes have at least 2 faces painted?

Accepted Answer

Out of 6 faces of 3 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 1 Number of vertices with 2 faces exposed (Painted) is 3 Number of vertices with 1 faces exposed (Painted) is 3 Number of vertices with 0 faces exposed (Painted) is 1 Number of sides with 2 sides exposed (Painted) is 3 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 3 From the above observation Number of cubes with 3 faces Painted is 1 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18 Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 1- 18 - 108 = 216 In other words number of cubes with 0 painted is (7 - 1)3 = 216. From the above explanation number of the cubes with at least 2 faces painted is 18 + 1 = 19.

Question 17

How many of the cubes have 3 faces painted?

Accepted Answer

Out of 6 faces of 3 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 1 Number of vertices with 2 faces exposed (Painted) is 3 Number of vertices with 1 faces exposed (Painted) is 3 Number of vertices with 0 faces exposed (Painted) is 1 Number of sides with 2 sides exposed (Painted) is 3 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 3 From the above observation Number of cubes with 3 faces Painted is 1 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18 Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 1- 18 - 108 = 216 In other words number of cubes with 0 painted is (7 - 1)3 = 216. From the above explanation number of the cubes with 3 faces painted is 1.

Question 18

How many of the cubes have 0 faces painted?

Accepted Answer

Out of 6 faces of 4 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 2 Number of vertices with 2 faces exposed (Painted) is 4 Number of vertices with 1 faces exposed (Painted) is 2 Number of vertices with 0 faces exposed (Painted) is 0 Number of sides with 2 sides exposed (Painted) is 5 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 1 From the above observation: Number of cubes with 3 faces Painted is 2 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and required number of cubes is 6 x 4 + 1 x 5 = 29 since there are 4 edges will give us 6 cubes from 1 edge and 1 edge (between two vertices which are painted or exposed from 3 sides) will give us only 5 cubes. Number of cubes with 1 face Painted is given by faces which is exposed from one sides and required number of cubes is 36 x 2 + 30 x 2 = 132 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 2 - 29 - 132 = 180 In other words number of cubes with 0 painted is 6 x 6 x 5 =180 From the above explanation number of the cubes with 0 faces painted is 180.

Question 19

How many of the cubes have 2 faces painted?

Accepted Answer

Out of 6 faces of 4 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 2 Number of vertices with 2 faces exposed (Painted) is 4 Number of vertices with 1 faces exposed (Painted) is 2 Number of vertices with 0 faces exposed (Painted) is 0 Number of sides with 2 sides exposed (Painted) is 5 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 1 From the above observation: Number of cubes with 3 faces Painted is 2 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and required number of cubes is 6 x 4 + 1 x 5 = 29 since there are 4 edges will give us 6 cubes from 1 edge and 1 edge (between two vertices which are painted or exposed from 3 sides) will give us only 5 cubes. Number of cubes with 1 face Painted is given by faces which is exposed from one sides and required number of cubes is 36 x 2 + 30 x 2 = 132 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 2 - 29 - 132 = 180 In other words number of cubes with 0 painted is 6 x 6 x 5 =180 From the above explanation number of the cubes with 2 faces painted is 29.

Question 20

How many of the cubes have at most faces painted?

Accepted Answer

Out of 6 faces of 4 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 2 Number of vertices with 2 faces exposed (Painted) is 4 Number of vertices with 1 faces exposed (Painted) is 2 Number of vertices with 0 faces exposed (Painted) is 0 Number of sides with 2 sides exposed (Painted) is 5 Number of sides with 1 sides exposed (Painted) is 6 Number of sides with no sides exposed (Painted) is 1 From the above observation: Number of cubes with 3 faces Painted is 2 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and required number of cubes is 6 x 4 + 1 x 5 = 29 since there are 4 edges will give us 6 cubes from 1 edge and 1 edge (between two vertices which are painted or exposed from 3 sides) will give us only 5 cubes. Number of cubes with 1 face Painted is given by faces which is exposed from one sides and required number of cubes is 36 x 2 + 30 x 2 = 132 Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 2 - 29 - 132 = 180 In other words number of cubes with 0 painted is 6 x 6 x 5 =180 From the above explanation number of the cubes with at most 2 faces painted is 180 + 132 + 29 = 341. Or else 343 - 2 = 341

Cubes and Dice Questions