Page 251 - Services Selection Board (SSB) Interviews

Cubes 247
Number of vertices with three faces exposed (Painted) is 2 Number of cubes with 0 face Painted is given by difference
Number of vertices with 2 faces exposed (Painted) is 4 between total number of cubes – number of cubes with at
Number of vertices with 1 faces exposed (Painted) is 2 least 1 face painted = 343 – 2 – 29 – 132 = 180
Number of vertices with 0 faces exposed (Painted) is 0 In other words number of cubes with 0 painted is
Number of sides with 2 sides exposed (Painted) is 5 6 × 6 × 5 = 180
Number of sides with 1 sides exposed (Painted) is 6 16. (d) From the above explanation number of the cubes
with 0 faces painted is 180.
Number of sides with no sides exposed (Painted) is 1
From the above observation: 17. (b) From the above explanation number of the cubes
with 2 faces painted is 29.
Number of cubes with 3 faces Painted is 2
Number of cubes with 2 faces Painted is given by sides 18. (a) From the above explanation number of the cubes
which is exposed from two sides and required number of with at most 2 faces painted is
cubes is 6 × 4 + 1 × 5 = 29 since there are 4 edges will 180 + 132 + 29 = 341.
give us 6 cubes from 1 edge and 1 edge (between two Or else 343 -2 = 341
vertices which are painted or exposed from 3 sides) will 19. (a) From the above explanation number of the cubes
give us only 5 cubes. with at least 2 faces painted is 29 + 2 = 31.
Number of cubes with 1 face Painted is given by faces 20. (d) From the above explanation number of the cubes
which is exposed from one sides and required number of with 3 faces painted is 2.
cubes is 36 × 2 + 30 × 2 = 132
Concept Builder

Solution for 1–5: In other words number of cubes with 0 painted is
Out of 6 faces of 5 faces are exposed and those were 6 × 5 × 5 = 150
painted. 1. (b) From the above explanation number of the cubes

Number of vertices with three faces exposed (Painted) is 4 with 0 faces painted is 150.
Number of vertices with 2 faces exposed (Painted) is 4 2. (c) From the above explanation number of the cubes
Number of vertices with 1 faces exposed (Painted) is 0 with 2 faces painted is 44.
Number of vertices with 0 faces exposed (Painted) is 0 3. (a) From the above explanation number of the cubes
Number of sides with 2 sides exposed (Painted) is 8 with at most 2 faces painted is
Number of sides with 1 sides exposed (Painted) is 4 150 + 145 + 44 = 339.
Or else 343 – 4 = 339
Number of sides with no sides exposed (Painted) is 0 4. (a) From the above explanation number of the cubes
From the above observation:
with at least 2 faces painted is 44 + 4 = 48.
Number of cubes with 3 faces Painted is 4 5. (d) From the above explanation number of the cubes
Number of cubes with 2 faces Painted is given by sides with 3 faces painted is 4.
which is exposed from two sides, out of 8 such edges 4
vertical edges will give us 6 cubes per edge and 4 edges Solution for 6–10:
from top surface will give us 5 such cubes from each edge Let us see the changes due to removal of 1 cube from
and required number of cubes is 6 × 4 + 4 × 5 = 44. corner-
Number of cubes with 1 face Painted is given by faces Number of vertices with three faces exposed (Painted) is
which is exposed from one sides four vertical faces will 7 + 3 = 10
give us 6 × 5 = 30 cubes per face and top face will give Number of Cubes with 2 sides exposed (Painted): In
us 5 × 5 = 25 and required number of cubes is 30 × 4 general one edge give us 4 (n – 2 in general case) cubes
+ 25 × 1 = 145 with two face painted but in this case out of 12 edges only
Number of cubes with 0 face Painted is given by difference 9 edges will give us 4 cubes in one edge and remaining
between total number of cubes – number of cubes with at 3 edges will give us 3 cubes from one edge, hence total
least 1 face painted = 343 – 4 – 44 – 145 = 150 number of edge is 9 × 4 + 3 × 3 = 45