Page 252 - Services Selection Board (SSB) Interviews

248 Cubes
Number of Cubes with 1 side exposed (Painted): It will 15. (b) In this case we have to use red and green twice
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remain same as normal case i.e 6(4 ) = 96 and same colour should be on opposite faces then
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Number of Cubes with no sides exposed (Painted) is 4 required cube is given by 4 edges (but not corner),
= 64 maximum number of cubes from one edge is 6 – 2
From the above observation: = 4 so required number of cubes is 4 × 4 = 16
6. (a) From the above explanation number of the cubes Solution for 16–20:
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with 0 faces painted is 64. 16. (c) Total no. of cubes = 5 = 125,
7. (c) From the above explanation number of the cubes Some cubes from different corners are removed and
with 2 faces painted is 45. the number removed cubes are 2, 3, 4 and 4.
8. (a) From the above explanation number of the cubes Remaining number of small cubes:
with at most 2 faces painted is 64 + 96 + 45 = 205. = 125 – 2 – 3 – 4 – 4 = 125 – 13 = 112
Or else 215 – 10 = 205 17. (c) In any plane,leave 4 sides cube and select (3 ×
9. (a) From the above explanation number of the cubes 3 × 3) inter section .But the cubes 2 × 2 ×1 give 2
with at least 2 faces painted is 45 + 10 = 55. less cube because that part we are already removed.
10. (d) No cubes are with 4 face painted. No. of cubes = (3 × 3 × 3) – 2 = 25.
Solution for 11–15: 18. (a) Only two faces are coloured is when cubes are at
For least number of cuts 120 = 4 × 5 × 6 i.e number the edges (baring the corner cubes)
of cuts must be 3, 4 and 5 in three planes in this case If no cubes have been removed then on each edges
number of cubes on a face is either 6 × 5 = 30 or 6 × 4 we will get 3 cubes that has exactly 2 faces coloured,
= 24 or 4 × 5 = 20 cubes. And number of cuboids on an hence total number of such cubes = 12 × 3 = 36,
edge is 4 or 5 or 6 because we have 12 edges.
11. (a) Number of cuboids with no face painted is (4 – 2) Out of these 3 cubes are removed hence required
(5 – 2)(6 – 2) = 2 × 3 × 4 = 24 number of cubes = 36 – 3 = 33
12. (a) To satisfy this case all the cuboids on the edges 19. (b) Each has Red faces on top layer = all edges cube
and corners must have more than one colour on = 2 + 2 + 2 + 2 = 8
them. And in that case opposite faces must have 20. (c) Number of cubes with 3 face coloured = 4
painted in the same colour. (Bottom cubes) + 8 top cubes + 4 (column cubes)
In that case number of cuboids with 3 colours on = 16
them = 8 Solution for 21–25:
In that case number of cuboids with 2 colours on Initial total number of cubes = 343,
them = 4 × (2 + 3 + 4) = 36
Hence number of cuboids with at least 1 colour on Number of cubes removed = 27
them is 120 – 36 – 8 = 76 Smaller 27 cubes painted blue
13. (c) In this case when k is maximum, one particular Exposed faces of original big cube (3 faces with 9 cube on
colour is used on three faces such that any two faces each face i.e total 27 cubes) painted with black
are adjacent to each other. Required number of 21. (c) Since 7 corner (Vertices) of bigger cube is
cuboids will come from edges but not from vertex = untouched hence they are painted with three faces.
3 + 4 + 5 + 1 = 13 Now consider the corner from where we have
14. (b) Maximum number of cuboid with red colour is removed 3 × 3 × 3 cubes,
possible when cube is painted with red colour in After removal 3 new corners of the bigger cube will
3 sides with minimum number of common edges be generated that will be painted with 3 faces and 8
(which is equal to 2) corners from smaller cube of 3 × 3 × 3 painted with
Hence required maximum value is 6 (5 + 5 + 4 – 2) 3 faces.
= 72 So the such total number of such cubes is 7 + 3 + 8
For minimum number of such cuboid Red colour is = 18.
used only once and minimum number of cubes in 22. (b) In original big cube number of faces with one
that case is 20 colour is 3(6 – 2) = 48 (here we have considered
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Hence required ratio is 72: 20 = 18: 5 only 3 untouched faces of big cube)