Q.1. the correct figure, when the given unfolded hollow cube is folded to make a perfect cube.

A. 1

B. 2

C. 3

D. 4

Sol : Option C In the first option, two blank sides & a side having star are drawn together, however this can’t be possible looking at the figure given in the question. Therefore, first option is incorrect. For this very reason, option two is also incorrect. The diagonally half filled side can not be next to both of the blank sides. Coming to option 3, looking at the question figure, full colored side, blank side & half covered diagonal side are three adjacent sides should be placed adjacent to each other. So it is the correct Sol. Option D is eliminated because when the diagonally half filled side is kept on side & laterally half-filled side is kept on top, the side visible to the right side should be the dotted side & not the blank side. Therefore, option D is eliminated. What is on the other side? Very often the puzzles & questions are related to what can be felt & seen. However, these types of puzzles are somewhat tricky. You are shown a few sides of the cube, & asked to tell about the opposite side of a particular face of the cube. For these types of puzzles, you need to imagine that you are rotating the cube. Refer to the example given below.

Q.2. In the given figure what will be on the surface opposite of F?

A. A

B. B

C. C

D. D

Sol : Option C In this puzzle, the same hollow cube is shown from different positions resulting in various cube figures. From the first two figures, it is clear that A and C are adjacent to each other. If we consider these two faces as lateral faces , let us assume that D forms the top side with A on S2 & C on S1. Now, if we invert this hollow cube, with the same two lateral sides facing towards us, Then C shifts to S2 & A shifts on to S1. Now, the bottom side of the cube becomes the top side of this cube. Now, we can infer that E becomes the bottom side of this cube. In the third figure, now if we place the top side of this hollow cube upright again, D becomes the top position in this cube. Now, it is clear that both the sides that face us that is S1 & S2 have changed. Now, we have F on S1 and B on S2 . So, A must have been rotated to S4 while C must have been rotated to S3. Thus, the symbol opposite to the F (on S1), is the symbol that is on S3, which is clearly C. Therefore, we can see how the first three figures are sufficient to Sol this question.

Q.3. Which symbol will be opposite to the face containing the symbol ?

A. !

B. Δ

C. ?

D. ÷

Sol : Option D The symbols on the adjacent faces of the cube to the face with symbols are !,o,? and ∆. Hence, the required symbol is ÷. Therefore, option D is the correct Sol.

Q.4. Two of the positions of dice are represented in the figure below. Which of the faces will be opposite to the face with 5?

A. 1

B. 2

C. 3

D. None of the above

Sol : Option D In these two positions given, one of the common faces to both the figures having 1 is in the exactly same position. Hence, according to rule, there must be 4 on the required face.

Q.5. Which of the following numbers will we find on the face opposite to the face containing the number 2?

A. 5

B. 4

C. 6

D. 1 or 4

Sol : Option C As discussed in the previous question, with the help of same rule, the number opposite to 2 should be number 6.

Q.6. Two different positions of an unbiased dice are shown in the figure below. Which digit should appear on the face opposite to the face with the digit 3?

A. 2 or 5

B. 6

C. 2

D. 3

Sol : Option D In this case, it is very obvious to see that 1, 2, 5 & 6 are adjacent to the digit 4. So the only number that can be opposite to 4 is 3. Hence option D, is the correct Sol.

Q.7. Which face is opposite to the face which contains 3?

A. 1

B. 5

C. 4

D. 6

Sol : Option C In first two positions of the dice, one common face with 5 is same. So, according to the rule no. (3), the face opposite to the face which contains 3, will contain 4. Here the common faces with number 3 are in same positions. Hence 6 is opposite to 2 and 5 is opposite to 1.

Directions for Questions (8-10): The sheet of paper shown in the figure (A) given on the leftmost side, in each question, is folded to form a box. Choose from amongst the alternatives (1), (2), (3) and (4), the boxes that are similar to the box that will be formed.

Q8. Which option has boxes that are similar to the box made from the given sheet of paper (A).

A. 2 & 3 only

B. 1,2 & 3 only

C. 1,2,3 & 4

D. 1 only

Sol : Option C When the sheet in the fig A is folded to form a hollow box, then the face that features ? is opposite to the face which is blank, the another blank face is opposite to the face that features with a ‘!’ sign, and the third blank face is opposite to the face with a triangle on it. Therefore, all the hollow cubes shown in all the four figures are possible.

Q9. Which option has boxes that are similar to the box made from the given sheet of paper(X).

A. 1, 3 and 4 only

B. 1 and 3 only

C. 1 only

D. 1, 2, 3 and 4

Sol : Option C When the sheet of paper shown in the fig (X) is folded to form a hollow cube, the blank faces are adjacent to each other. The face featuring a ! in it is opposite to another blank face & the face with the ? opposite to the o in the figure. Hence, only the hollow cube shown in the figure 1 is possible.

Q10. Which option has boxes that are similar to the box made from the given sheet of paper(X)

A. 1, 2, 3 and 4

B. 1 and 2 only

C. 3 and 4 only

D. 1 and 3 only

Sol : Option D When a hollow cube is made by folding the sheet of paper shown in fig. (X), then & are the two faces of the cube which are opposite to each other. Also, the face which shows the black circle sign lies opposite to the face with the sign 'x' & the face featuring the square lies opposite to the face with the white circle. Now, the hollow cubes in the figure (4) and figure (2) consist of the faces which are not at all possible when the sheet in the fig. (X) is folded to form a hollow cube. Hence, these two hollow cubes are not possible. Hence, only the hollow cubes in figure (1) and figure (3) are possible..