Question 1

How many of the cubes have at least 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 at least 2 faces painted is 29 + 2 = 31.

Question 2

How many of the cubes have 3 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 3 faces painted is 2.

Question 3

How many of the cubes have 0 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.

Question 4

How many of the cubes have 2 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 2 faces painted is 44.

Question 5

How many of the cubes have at least 2 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 least 2 faces painted is 44 + 4 = 48.

Question 6

How many of the cubes have 0 faces painted?

Accepted Answer

Let us see the changes due to removal of cube from corner- Number of vertices with three faces exposed (Painted) is 7 + 3 = 10 Number of Cubes with 2 sides exposed (Painted): In general one edge gives us 4 (n - 2 in general case) cubes with two face painted but in this case out of 12 edges only 9 edges will give us 4 cubes in one edge and remaining 3 edges will give us 3 cubes from one edge, hence total number of edge is 9 x 4 + 3 x 3 = 45 Number of Cubes with 1 side exposed (Painted): It will remain same as normal case i.e. 6(42) = 96 Number of Cubes with no sides exposed (Painted) is 43 = 64 From the above observation: From the above explanation number of the cubes with 0 faces painted is 64.

Question 7

How many of the cubes have 2 faces painted?

Accepted Answer

Let us see the changes due to removal of cube from corner- Number of vertices with three faces exposed (Painted) is 7 + 3 = 10 Number of Cubes with 2 sides exposed (Painted): In general one edge gives us 4 (n - 2 in general case) cubes with two face painted but in this case out of 12 edges only 9 edges will give us 4 cubes in one edge and remaining 3 edges will give us 3 cubes from one edge, hence total number of edge is 9 x 4 + 3 x 3 = 45 Number of Cubes with 1 side exposed (Painted): It will remain same as normal case i.e. 6(42) = 96 Number of Cubes with no sides exposed (Painted) is 43 = 64 From the above observation: From the above explanation number of the cubes with 2 faces painted is 45.

Question 8

How many of the cubes have at most faces painted?

Accepted Answer

Let us see the changes due to removal of cube from corner- Number of vertices with three faces exposed (Painted) is 7 + 3 = 10 Number of Cubes with 2 sides exposed (Painted): In general one edge gives us 4 (n - 2 in general case) cubes with two face painted but in this case out of 12 edges only 9 edges will give us 4 cubes in one edge and remaining 3 edges will give us 3 cubes from one edge, hence total number of edge is 9 x 4 + 3 x 3 = 45 Number of Cubes with 1 side exposed (Painted): It will remain same as normal case i.e. 6(42) = 96 Number of Cubes with no sides exposed (Painted) is 43 = 64 From the above observation: From the above explanation number of the cubes with at most 2 faces painted is 64 + 96 + 45 = 205. Or else 215 - 10 = 205

Question 9

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

Accepted Answer

Let us see the changes due to removal of cube from corner- Number of vertices with three faces exposed (Painted) is 7 + 3 = 10 Number of Cubes with 2 sides exposed (Painted): In general one edge gives us 4 (n - 2 in general case) cubes with two face painted but in this case out of 12 edges only 9 edges will give us 4 cubes in one edge and remaining 3 edges will give us 3 cubes from one edge, hence total number of edge is 9 x 4 + 3 x 3 = 45 Number of Cubes with 1 side exposed (Painted): It will remain same as normal case i.e. 6(42) = 96 Number of Cubes with no sides exposed (Painted) is 43 = 64 From the above observation: From the above explanation number of the cubes with at least 2 faces painted is 45 + 10 = 55.

Question 10

How many of the cubes have exactly 4 face painted?

Accepted Answer

Let us see the changes due to removal of cube from corner- Number of vertices with three faces exposed (Painted) is 7 + 3 = 10 Number of Cubes with 2 sides exposed (Painted): In general one edge gives us 4 (n - 2 in general case) cubes with two face painted but in this case out of 12 edges only 9 edges will give us 4 cubes in one edge and remaining 3 edges will give us 3 cubes from one edge, hence total number of edge is 9 x 4 + 3 x 3 = 45 Number of Cubes with 1 side exposed (Painted): It will remain same as normal case i.e. 6(42) = 96 Number of Cubes with no sides exposed (Painted) is 43 = 64 From the above observation: No cubes are with 4 face painted.

Question 11

What is the number of cubes with no face painted?

Accepted Answer

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6 Number of cuboids with no face painted is (4 - 2)(5 - 2)(6 - 2) = 2 x 3 x 4 = 24

Question 12

What is the least possible number of piece which have at most one colour on them?

Accepted Answer

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6 To satisfy this case all the cuboids on the edges and corners must have more than one colour on them. And in that case opposite face must have painted in the same colour. In that case number of cuboids with 3 colours on them = 8 In that case number of cuboids with 2 colours on them = 4 x (2 + 3 + 4 ) = 36 Hence number of cuboids with at least 1 colour on them is 120 - 36 - 8 = 76

Question 13

If K is the number of cuboids which have more than one face painted in the same colour then find the maximum value of k.

Accepted Answer

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6 In this case when k is maximum, one particular colour is used on there faces such that any two faces are adjacent to each other. Required number of cuboids will come from edges but not from vertex = 3 + 4 + 5 + 1 = 13

Question 14

What is the ratio of maximum and minimum number of cuboids with red colour on them?

Accepted Answer

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6 Maximum number of cuboid with red colour is possible when cube is painted with red colour in 3 sides with minimum number of common edges (which is equal to 2) Hence required maximum value is 6 (5 + 5 + 4 - 2) = 72 For minimum number of such cuboid Red colour is used only once and minimum number of cubes in that case is 20 Hence required ratio is 72 : 20 = 18 : 5

Question 15

What is the maximum number of cuboids with only two face painted one face is painted red and other green?

Accepted Answer

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6 In this case we have to use red and green twice and same colour should be on opposite faces then required cube is given by 4 edges (but not corner), maximum number of cubes one edge is 6 - 2 = 4 so required number of cubes is 4 x 4 = 16

Question 16

How many small cubes are left in the block?

Accepted Answer

Total no. of cubes = 53 = 125, Some cubes from different corners are removed and the number removed cubes are 2, 3, 4 and 4. Remaining number of small cubes: = 125 - 2 - 3 - 4 - 4 = 125 - 13 = 112

Question 17

How many cubes do not have any coloured face?

Accepted Answer

In any plane, leave 4 sides cube and select (3 x 3 x 3) inter section. But the cubes 2 x 2 x 1 give 2 less cube because that part we are already removed. No. of cubes = (3 x 3 x 3) - 2 = 25.

Question 18

How many cubes have only two coloured faces?

Accepted Answer

Only two faces are coloured is when cubes are at the edges (baring the corner cubes) If no cubes have been removed then on each edges we will get 3 cubes that has exactly 2 faces coloured, hence total number of such cubes = 12 x 3 = 36, because we have 12 edges. Out of these 3 cubes are removed hence required number of cubes = 36 - 3 = 33

Question 19

How many cubes in the top layer have three red faces each?

Accepted Answer

Each has Red faces on top layer = all edges cube = 2 + 2 + 2 + 2 = 8

Question 20

How many cubes with 3 facefacecoloured?

Accepted Answer

Number of cubes with 3 face coloured = 4 (Bottom cubes) + 8 top cubes + 4 (column cubes) = 16

Cubes and Dice Questions