Question 1

What is the number of small cubes which have exactly three faces painted?

Accepted Answer

Initial total number of cubes = 343, Number of cubes removed = 27 Smaller 27 cubes painted blue Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black Since 7 corner (Vertices) of bigger cube is untouched hence they are painted with three faces. Now consider the corner from where we have removed 3 x 3 x 3 cubes, After removed 3 new corners of the bigger cube will be generated that will be painted with 3 faces and 8 corners from smaller cube of 3 x 3 x 3 painted with 3 faces. So the such total number of such cubes is 7 + 3 + 8 = 18.

Question 2

What is the number of small cubes with exactly one face painted?

Accepted Answer

Initial total number of cubes = 343, Number of cubes removed = 27 Smaller 27 cubes painted blue Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black In original big cube number of faces with one colour is 3(6 -2)2 = 48 (here we have considered only 3 untouched of big cube) But here we have removed a cubes of the form of 3 x 3 x 3 and again put it back so out of three new exposed faces of big cube we will have 4 cubes in each face that is painted with one colour hence number of cubes from these three surfaces is 3 x 4 = 12 Now consider out of 3 x 3 x 3 cubes we will have 6 cubes (one in each face ) which are painted only one face. Hence total number of cubes = 48 + 12 + 6 = 66

Question 3

What is the number of small cube with no face painted

Accepted Answer

Initial total number of cubes = 343, Number of cubes removed = 27 Smaller 27 cubes painted blue Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black Without any changes number of cubes with no face colour is given by (6 - 2)3 = 64 Now because of removal of 3 x 3 x 3 cubes from one of the corner from each face that were not painted earlier got exposed and will get painted, so from 3 x 3 x 3 cubes 4 x 3 = 12 cubes got painted, and a similar number from 3 exposed faces of big cube got painted. Total number of cubes with no face painted is 64 - 12 - 12 = 40

Question 4

How many cubes are painted with yellow and blue?

Accepted Answer

Initial total number of cubes = 343, Number of cubes removed = 27 Smaller 27 cubes painted blue Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black Out of 27 small cubes from 3 x 3 x 3, outer 26 cubes are 1st painted with blue and then it is kept back with original cube and painted with yellow so out of 26 cubes only 5 edges will give us cubes with both the colours and number of such cubes are 12

Question 5

How many cubes are painted two faces only one with yellow and one with blue?

Accepted Answer

Initial total number of cubes = 343, Number of cubes removed = 27 Smaller 27 cubes painted blue Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black Out of 12 cubes in previous question there are 4 cubes with 2 faces yellow so number of cubes painted two faces only one with yellow and one with blue is 12 - 4 = 8

Question 6

How many small cubes are left in the block?

Accepted Answer

Number of cubes removed from top face = 16 Number of cubes removed from bottom face = 4 Number of cubes left = 125 - (16 + 4 ) = 105

Question 7

How many cubes are with one face painted?

Accepted Answer

From top face (out of 3 x 3 square face) only one cubes is with one face painted. From 4 vertical faces each face will give us 6 cubes hence total number of cubes from vertical faces is 6 x 4 = 24. From bottom face we will get 3 x 3 = 9 cubes So total number of cubes with one face painted is 1 + 24 + 9 = 34

Question 8

How many cubes are with no face painted?

Accepted Answer

Number of cubes with no face painted is 105 - 34 - 24 - 20 = 27 Or else all the 3 x 3 x 3 inner cubes will remain coloured.

Question 9

How many cubes have three red faces each?

Accepted Answer

Number of cubes with three coloured face on the top side = 4 Number of cubes with three coloured face on the 2nd from top side = 4 Number of cubes with three coloured face on the bottom side = 12 Total number of such cubes = 12 + 8 = 20

Question 10

How many cubes are with two faces painted?

Accepted Answer

Number of cubes with two face painted from the top side (Which is a square of 3 x 3 = 9 cubes ) is 4. Number of cubes with two face painted from the 2nd from top side (Which has four edges and edge has 3 such cubes) is 4 x 3 = 12. Number of such cubes from vertical edges is 4 x 1 = 4 Number of such cubes from bottom face is 4 x 1 = 4 Hence total such cubes is 4 + 12 + 4 + 4 = 24

Question 11

How many cubes are painted with red or blue?

Accepted Answer

Here on each face 6 x 6 = 36 cubes that are painted with one colour. Case (i): When red and blue are adjacent to each other then from one face we will get 6 x 6 = 36 cubes but out of them 6 cubes from common edge is common so number of cubes are 2 x 6 x 6 - 6 = 66 Case (ii): When red and blue are opposite to each other then required number of cubes is 2 x 6 x 6 = 72

Question 12

How many cubes are painted with red or blue or green?

Accepted Answer

Here on each face 6 x 6 = 36 cubes that are painted with one colour. Case (i): when these three colour are adjacent to each other then from one face we will get 6 x 6 = 36 cubes but out them 6 x 3 = 18 cubes from common edge is common so number of cubes are 3 x 6 x 6 - 6 x 3 = 90 Case (ii): When red and blue are opposite to each other (or any two of the given three) then required number of cubes is 3 x 6 x 6 - 2 x 6 = 96

Question 13

How many cubes are painted with red or blue but not green?

Accepted Answer

Here on each face 6 x 6 = 36 cubes that are painted with one colour. Case (i): When red and blue are opposite to each other then from one face we will get 6 x 6 = 36 cubes bot out of them 2 x 6 cubes from common edge with green painted face is common so number of cubes are 2 x 6 x 6 - 2 x 6 = 60 Case (ii): When red and blue are adjacent to each other then green is either adjacent to these or opposite to any one of red or blue, in 1st condition number of cubes is 2 x 6 x 6 - 2 x 6 - 11 = 55 cubes or in 2nd condition 2 x 6 x 6 -6 - 6 = 60, required number of cubes is 55 or 60

Question 14

Which of the following statement is correct (i) At least 1 cube is painted with red, green and blue. (ii) At most 1 cube is painted with red, green and blue. (iii) At most 6 cubes are painted with red and green. (iv) At least 6 cubes are painted wit…

Accepted Answer

Here on each face 6 x 6 = 36 cubes that are painted with one colour. From solution of previous questions statements (ii) and (iii) are correct.

Question 15

How many cubes are painted with red, blue, green and black?

Accepted Answer

Here on each face 6 x 6 = 36 cubes that are painted with one colour. None of the cubes can be painted in four faces.

Question 16

How many of the cubes have at most faces painted?

Accepted Answer

Out of 6 faces of 5 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 4 Number of vertices with 2 faces exposed (Painted) is 4 Number of vertices with 1 faces exposed (Painted) is 0 Number of vertices with 0 faces exposed (Painted) is 0 Number of sides with 2 sides exposed (Painted) is 8 Number of sides with 1 sides exposed (Painted) is 4 Number of sides with no sides exposed (Painted) is 0 From the above observation: Number of cubes with 3 faces Painted is 4 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides, out of 8 such edges 4 vertical edges will give us 6 cubes per edge and 4 edges from top surface will give us 5 such cubes from each edge and required number of cubes is 6 x 4 + 4 x 5 = 44. Number of cubes with 1 face Painted is given by faces which is exposed from one sides four vertical faces will give us 6 x 5 = 30 cubes per face and top face will give us 5 x 5 = 25 and required number of cubes is 30 x 4 + 25 x 1 = 145 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 - 4 - 44 - 145 = 150 In other words number of cubes with 0 painted is 6 x 5 x 5 = 150 From the above explanation number of the cubes with 0 faces painted is 150. From the above explanation number of the cubes with at most 2 faces painted is 150 + 145 + 44 = 339. Or else 343 - 4 = 339

Question 17

How many of the cubes have 3 faces painted?

Accepted Answer

Out of 6 faces of 5 faces are exposed and those were painted. Number of vertices with three faces exposed (Painted) is 4 Number of vertices with 2 faces exposed (Painted) is 4 Number of vertices with 1 faces exposed (Painted) is 0 Number of vertices with 0 faces exposed (Painted) is 0 Number of sides with 2 sides exposed (Painted) is 8 Number of sides with 1 sides exposed (Painted) is 4 Number of sides with no sides exposed (Painted) is 0 From the above observation: Number of cubes with 3 faces Painted is 4 Number of cubes with 2 faces Painted is given by sides which is exposed from two sides, out of 8 such edges 4 vertical edges will give us 6 cubes per edge and 4 edges from top surface will give us 5 such cubes from each edge and required number of cubes is 6 x 4 + 4 x 5 = 44. Number of cubes with 1 face Painted is given by faces which is exposed from one sides four vertical faces will give us 6 x 5 = 30 cubes per face and top face will give us 5 x 5 = 25 and required number of cubes is 30 x 4 + 25 x 1 = 145 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 - 4 - 44 - 145 = 150 In other words number of cubes with 0 painted is 6 x 5 x 5 = 150 From the above explanation number of the cubes with 0 faces painted is 150. From the above explanation number of the cubes with 3 faces painted is 4.

Question 18

If number of cubes painted with colour 3 is 217 then how many cubes are painted with colour5

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. From the given condition Number of cubes removed in 3rd step (i.e with colour 3) is = 3(N - 2)2 - 3(N - 2) + 1 = 217 hence N = 11 So number of cubes with colour 5 is = 3(N - 4)2 - 3(N - 4) + 1 = 127

Question 19

If after 7 th step number of cubes painted in exactly 2 faces with colour 7 is 21, then what is the number of cubes removed in 3 rd step.

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. Total number of Cubes left after 7thstep in (N - 7)3 in the form of (N - 7) x (N - 7) x (N - 7) cubes. And out of these number of cubes whose two sides are painted is given by three edges with each edge has (N - 8) so total number of cubes is 3 x (N - 8) From the given information 3(N - 8) = 21 or N = 15 Number of cubes removed in 3rd step (i.e with colour 3) is = 3(N - 2)2 - 3(N - 2) + 1 = 469

Question 20

Which of the following can be the number of cubes removed from the original N 3 number of cubes. (i) 37 (ii) 61 (iii) 98

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. The required number of cubes must be equal to difference between two positive integer Since 64 - 27 = 37 125 - 27 = 98 125 - 64 = 61

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