HackerRank Cubes and Dice Quiz -1

When the given sheet in the figure (X) is folded to form a hollow cube, the face with dot lies opposite to one of shaded faces. So, the hollow cube shown in the figure (2) which has both the shaded faces that are adjacent to the face with a dot, cannot be formed in any case. Therefore, the hollow cubes shown in figure (1), figure (3) and figure(4) can be formed.

Here we can see there are 3 rows and 4 columns of cubes. Therefore, number of cubes = 4*3 = 12

Again, another set of 4 cubes are there on top of the first row. Hence total number of cubes = 12 + 4 = 16

Number of small cubes having only two faces coloured = 6 from the front + 6 from the back + 2 from the left + 2 from the right = 16.

Such cubes are related to the corners of the cuboid and there are 8 corners.Hence, the required number is 8

Number of small cubes that have no face coloured = (4 - 2) x (3 - 2) = 2 x 1 = 2.

Number of small cubes having only one face coloured = 2 x 2 + 2 x 2 + 2 x 1 = 4 + 4 + 2 = 10

A 6 × 6 × 6 has 6 cubes of 1 × 1 × 1 along its length, breadth and width. If we add a layer of 1 × 1 × 1 cubes, then the new cube will have the following dimensions:

Length = 6 + 1 + 1 = 8; Breadth = 6 + 1 + 1 = 8; Width = 6 + 1 + 1 = 8

Therefore, the number of  1× 1 × 1 cubes in this = 8 × 8 × 8 = 512

Common faces with number 3, are in the same positions. Hence when the digits 4 is on the bottom then 1 will be on the upper surface. Face 4 is opposite 1, and face 6 is opposite face

If 1 is adjacent to 2, 3 and 5, then either 4 or 6 lies opposite to 1. So, the numbers 4 and 6 cannot lie opposite to each other. Hence, 4 necessarily lies adjacent to 6.

In a usual dice, the sum of the numbers on any two opposite faces is always 7. Thus, 1 is opposite 6, 2 is opposite 5 and 3 is opposite 4.

Consequently, when 4, 3, 1 and 5 are the numbers on the top faces, then 3, 4, 6 and 2 respectively are the numbers on the face touching the ground. The total of these numbers = 3 + 4 + 6 + 2 = 15.