Question 1

If number of cubes painted on exactly one face with colour 1 and colour 2 is 150, then how many cubes are painted with only colour 4?

Accepted Answer

Consider the 1st step, initial number of cubes N3 after removal of 1st set of coloured cubes number of cubes left out is (N - 1)3 hence number of cubes removed in 1st step (i.e with colour 1) is N3 - (N - 1)3 = 3N2 - 3N + 1 Similarly number of cubes removed in 2nd step (i.e with colour 2) is Similarly number of cubes removed in 3rd step is (i.e with colour 3) and so on. = 3(N - 1)2 - 3(N - 1) + 1 Number of cubes remaining after 1st step is (N - 1)3 Number of cubes remaining after 2nd step is (N - 2)3 and so on. Number of cubes with only face is painted with colour 1 is 3(N - 2)(N - 1) = 3N2 - 9N + 6 Number of cubes with only face is painted with colour 2 is 3 (N - 3)(N - 2) = 3N2 - 15 N + 18 From the given condition (3N2 - 9N + 6) + (3N2 - 15 N + 18) = 6N2 - 24N + 24 = 150 from this we will get N = 7. Number of cubes left after step 3 is 4 x 4 x 4 = 64 When all the exposed faces are painted with colour 4 then number of cubes with only one face painted is 3 x 2 x 3 = 18 From the observation for 1st step we have seen that number of cubes is 3(N - 2)(N - 1) or in other words 3 times the product of 2 consecutive integer that is satisfied only by 18 which is 3 times of 2 x 3.

Question 2

Of all the removed cubes which one of the following could be the number of cubes with exactly 2 face painted after 3 steps?

Accepted Answer

Consider the 1st step, initial number of cubes N3 after removal of 1st set of coloured cubes number of cubes left out is (N - 1)3 hence number of cubes removed in 1st step (i.e with colour 1) is N3 - (N - 1)3 = 3N2 - 3N + 1 Similarly number of cubes removed in 2nd step (i.e with colour 2) is Similarly number of cubes removed in 3rd step is (i.e with colour 3) and so on. = 3(N - 1)2 - 3(N - 1) + 1 Number of cubes remaining after 1st step is (N - 1)3 Number of cubes remaining after 2nd step is (N - 2)3 and so on. After step 1 number of cubes with exactly 2 face painted is 4(N - 1) + (N - 2) = 5N - 6 Similarly after 2nd step number of cubes with exactly 2 face painted is 5(N - 2) - 6 = 5N - 11 And after 3rd step number of cubes with exactly 2 face painted is 5(N - 2) - 6 = 5N - 16 So total number of such cubes is 15N - 33 out of the given options only option B satisfy the given condition.

Question 3

How many cubes are there in below figure?

Accepted Answer

There is 6 columns with 2 cubes each. Total boxes = 6 × 2 = 12

Question 4

How many cubes are there in below figure?

Accepted Answer

There are total 4 rows and 5 boxes in each row, then 5 × 4 = 20 boxes.

Question 5

How many cubes are there in below figure?

Accepted Answer

4 columns with 1 cubes = 4 cubes 1 columns with 3 cubes = 3 1 columns with 4 cubes = 4 Total cubes = 4 + 3 + 4 = 11

Question 6

Two Position of a dice are shown below as (i) and (i). When six is at the bottom, what number will be at the top?

Accepted Answer

From both the figures we find that numbers 1,2 and 4 dots appear adjacent to 6. thus, the number 5 dots will appear opposite to 6. Therefore when 6 is at the bottom, then 5 will be at the top.

Question 7

Two positions of a dice are shown below (i) and (ii). When 1 is at the top. which number will be at the bottom?

Accepted Answer

as we can observe the position of 6 is same in both the figures and position of 4 moves from Right hand side to Left hand side (clock wise direction).

Question 8

How many dots are contained on the faces opposite to that containing four dots?

Accepted Answer

as both the figure contains one dot and 5 dots, fixing the 5 dots face and moving one dot face position to fig (i). we find 4 dots is opposite to 2 dots. For the better understanding take a dice and apply the same steps which is mentioned above.

Question 9

Below there are three different positions of a dice. Find the number of dots on the face opposite to the face with one dot.

Accepted Answer

as 1, 3 5 and 6 are adjacent to 2 in all the three positions, if 3 is opposite to 5 that we came across from the figures. So it is obvious that one will be opposite to 6. For the better understanding take a dice ans apply the same steps which is mentioned above.

Question 10

Among the four alternatives which one cannot be the similar box as the question figure.

Accepted Answer

as the dot face cannot be adjacent to both the dark shaded faces. Hence option c cannot be possible.

Question 11

Among the four alternatives which one cannot be the similar box as the question figure.

Accepted Answer

in option (b) "?" and "/" are adjacent to each other, which cannot be possible.

Question 12

Among the four alternatives choose which are can be the similar boxes as the question figure.

Accepted Answer

(i) and (ii) cannot be possible as the shaded face will not be adjacent to doted face.

Question 13

From the following two different appearance of a dice, which number lies to the opposite of 2?

Accepted Answer

According to the rule ( 1 ) of dice , we know that Two opposite faces cannot be adjacent to one another. Clearly, the number 4 lies to the opposite of 2.

Question 14

Two positions of a dice are given. Which number would be at the top when bottom is 2?

Accepted Answer

On the basis of question figure pattern , The numbers 1, 2, 5 and 6 are on the adjacent faces of number 3. Therefore, the number 4 lies opposite 3. The numbers 3, 4 and 6 cannot be on the faces opposite to 1. Therefore, 5 lies opposite 1. Now, 2 lies opposite 6.

Question 15

Four positions of a dice are given below. Identify the number at the bottom when top is 6.

Accepted Answer

On the basis of question figure pattern , we can see that The numbers 1, 2, 5 and 6 are on the faces adjacent to the number 4. So, the number 3 lies opposite 4. The numbers 1, 3, 4 and 6 lies on the faces adjacent to the number 2. So, the number 5 lies opposite 2. Now, the number 1 lies opposite 6.

Question 16

Four positions of a dice are given below. Identify the number at the bottom when the number on the top is 2.

Accepted Answer

On the basis of question figure pattern , The numbers 1, 2, 5 and 6 are on the faces adjacent to the number 3. Therefore, 4 lies opposite 3. The numbers 2, 3, 4 and 5 are on the faces adjacent to the number 1. Therefore, 6 lies opposite 1. Now, the number 5 lies opposite 2.

Question 17

Four positions of a dice are given below: Find the number on the face opposite to the face showing 4.

Accepted Answer

As we can see that , The numbers 1, 2, 3 and 6 lie on the faces adjacent to the number 5. Therefore, the number 5 lies opposite 4.

Question 18

Which colour is opposite to Green?

Accepted Answer

On the basis of question figure pattern , The six colours are: Indigo, Violet, Orange, Red, Green and Blue. The colours Indigo, Blue and Red are on the faces adjacent to Green and therefore, these colours cannot be on the face opposite to Green. After careful observation of all the views of dice, it is clear that Orange lies opposite to Green.

Question 19

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

Accepted Answer

As per the given pattern in question figure , we can find the similar next position of given dice . The numbers 1, 3, 4 and 5 are on adjacent faces of the number Therefore, 6 lies opposite 2.

Question 20

Study the two different positions of a dice. When the face containing one dot is at bottom then how many dots would be there on the top face?

Accepted Answer

As per the given pattern in question figure , we can find the similar next position of given dice . When both 2 and 3 dots are at the same side, 1 dot should be opposite to 4 dot.

Cubes and Dice Questions