Cubes & Dice MCQs for UPSC Prelims

Faces that appear together in a view share an edge, so they cannot be opposite. The two dots appear with the line and the single dot (View I), with the three dots (View II) and with the cross (View IV). The only symbol never seen next to two dots is four dots, so four dots is the opposite face. The single dot, three dots and the line are all adjacent to it.

Faces that appear together in a view share an edge, so they cannot be opposite. The cross appears with the single dot and four dots (View III) and with three dots and two dots (View IV). The only symbol never seen next to the cross is the line, so the line is the opposite face. The single dot, two dots and four dots are all adjacent to the cross.

List the neighbours of each colour. From clues 1, 2 and 3, Y is adjacent to O, B, I and G, so the face opposite Y is V. From clues 1 and 4, O is adjacent to Y, B and V. Since a face touches four others and is opposite the one it never shares an edge with, O cannot be opposite Y, B or V, so O is opposite either I or G. From clues 1, 3 and 4, B is adjacent to Y, O, G and V, leaving I as the face opposite B. That uses up I as B's opposite, so the only remaining colour to sit opposite O is G. Hence the face opposite O is coloured G. (The scanned OCR dropped the fourth clue; it is restored here from the original paper.)

A cube has three pairs of opposite faces. By tracking which symbols appear together (adjacent) across the rotated views, the faces that share an edge with the single dot can be eliminated as its opposite. Working through the rotations, the single dot is never seen adjacent to the three dots, while it does appear next to four dots, two dots, the cross and the line; the only face it can never sit beside is three dots, so three dots must be the hidden opposite face. Hence (b) Three dots is correct.

From the first two views, A, B, H and M all appear adjacent to K, so each of them shares an edge with K and none of them is opposite K. Since A is adjacent to B and to M, the face opposite A must be one of the remaining candidates, H or M. Testing the possibilities: if M were opposite A, then H would have to be opposite B, but the third view shows H and B configured in a way that makes this impossible. Therefore H is opposite A, answer (a).

The cuboid yields 7 x 5 x 3 = 105 unit cubes. Statement 1: unpainted cubes form the inner block (7-2) x (5-2) x (3-2) = 5 x 3 x 1 = 15, so it is correct. Statement 2: green sits on the 5 x 3 faces and blue on the 7 x 3 faces, so a cube showing exactly green and blue must lie on an edge where those faces meet - the four vertical edges of length 3. Each such edge has 3 cubes, but its two end cubes are corners that also touch a red 7 x 5 face (three painted faces), leaving 3 - 2 = 1 qualifying cube per edge, i.e. 4 cubes in total, not 6. Statement 2 is wrong. Hence option (a); (b) and (c) accept the miscount of 6, and (d) wrongly rejects the straightforward interior count of 15.

In a straight column of four squares, the first and third squares fold onto opposite faces, and so do the second and fourth. So Green is opposite Orange and Blue is opposite White. The two squares attached on either side of Blue (Yellow and Red) also end up opposite each other. Red, Green and Orange therefore all share an edge with White on the cube, and only Blue is opposite White.

Option (a) is correct. 125 identical cubes form a 5 x 5 x 5 block. A small cube is surrounded on every side exactly when it does not touch any outer face of the block, i.e. it lies in the inner core obtained by stripping one layer from each side. That core measures (5 - 2) x (5 - 2) x (5 - 2) = 3 x 3 x 3 = 27 cubes. Option (b) 25 is a face count, not a volume; options (c) 21 and (d) 18 correspond to no consistent counting of the hidden core.