Dice Cubes And Painted Faces MCQs

A 5 cm cube is painted on all faces and cut into 1 cm³ cubes. What percentage of the smaller cubes have no paint on any face?

32.4%
21.6%
17.6%
27.2%

Explanation: Total unit cubes = 5³ = 125. Interior unpainted cubes = (5 − 2)³ = 27. Percentage = (27 / 125) × 100 = 21.6%.

An 8 cm cube is painted on all faces and cut into 1 cm³ cubes. What is the ratio of the number of cubes with exactly one painted face to the number of cubes with exactly two painted faces?

4:1
2:1
6:1
3:1

Explanation: Using the formulas: cubes with exactly one painted face equal 6(n − 2)² = 6 × 36 = 216. Cubes with exactly two painted faces equal 12(n − 2) = 12 × 6 = 72. The ratio is 216 : 72 = 3 : 1.

A cube of side 5 cm is painted on all faces and then cut into 1 cm³ smaller cubes. How many smaller cubes have exactly two faces painted?

48
32
24
36

Explanation: For an n×n×n cube cut into unit cubes, the cubes with exactly two painted faces lie along the edges but exclude the corners. There are 12 edges, and each edge contributes (n − 2) such cubes. Using the formula: 12(n − 2) = 12 × 3 = 36.

A cube of side 10 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on at least one face?

448
488
784
512

Explanation: The total number of unit cubes is 10³ = 1000. The cubes with no paint form an inner cube of side (10 − 2) = 8, giving 8³ = 512 unpainted cubes. Subtracting: 1000 − 512 = 488 cubes have paint on at least one face.

A 5 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and are located on the vertical edges only (not on the top or bottom edges)?

16
24
12
8

Explanation: A cube has 4 vertical edges. Each vertical edge contains (n − 2) cubes with exactly two painted faces (excluding the corner cubes which have three painted faces). Therefore: 4 × (5 − 2) = 4 × 3 = 12.

A 10 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly three painted faces?

6
12
8
4

Explanation: Cubes with exactly three painted faces are always the 8 corner cubes of the original cube, regardless of the side length. Thus, there are always 8 such cubes.

A 2 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly one painted face?

4
2
6
0

Explanation: A 2 cm cube cut into 1 cm³ cubes yields 8 unit cubes, all of which are corner cubes. Since every unit cube occupies a corner position, each has three painted faces, leaving zero cubes with exactly one painted face.

A cuboid of dimensions 5 cm × 7 cm × 9 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on at least two faces?

60
68
52
76

Explanation: Cubes with at least two painted faces include the 8 corner cubes (three painted faces) and the non-corner edge cubes (two painted faces). Using the formulas: 8 + 4[(a − 2) + (b − 2) + (c − 2)] = 8 + 4 × 15 = 68.

A cuboid of dimensions 5 cm × 6 cm × 7 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two faces painted?

48
72
60
36

Explanation: For a cuboid, cubes with exactly two painted faces lie on the edges excluding corners. The formula is: 4[(a − 2) + (b − 2) + (c − 2)]. Substituting: 4[(5 − 2) + (6 − 2) + (7 − 2)] = 4 × 12 = 48.

A 4 cm cube is painted on exactly five faces, leaving one face unpainted. It is then cut into 1 cm³ cubes. How many smaller cubes have paint on exactly two faces?

16
12
24
20

Explanation: Assume the bottom face is unpainted (the answer is the same for any single unpainted face due to symmetry). The cubes with exactly two painted faces are: (i) the four non-corner edges of the top face, giving 4(n − 2) = 8; (ii) the four non-corner vertical edges, giving 4(n − 2) = 8; and (iii) the four corners of the unpainted bottom face, giving 4. Total: 8 + 8 + 4 = 20.

Two positions of the same die are shown. In the first position, faces 2, 3 and 5 are visible. In the second position, faces 2, 4 and 6 are visible. Which number is opposite to 2?

1
6
3
4

Explanation: From the first position, 2 is adjacent to 3 and 5, so it cannot be opposite either of them. From the second position, 2 is adjacent to 4 and 6, so it cannot be opposite either of them. The only remaining face is 1, which must be opposite to 2.

A cuboid of dimensions 6 cm × 8 cm × 10 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on at least one face?

312
288
336
192

Explanation: The total number of unit cubes is 6 × 8 × 10 = 480. The completely unpainted interior forms a cuboid of dimensions (6 − 2) × (8 − 2) × (10 − 2) = 4 × 6 × 8 = 192. Subtracting the unpainted from the total: 480 − 192 = 288.

Four positions of the same die are shown. In Position I, faces 1, 2 and 3 are visible. In Position II, faces 1, 2 and 4 are visible. In Position III, faces 1, 3 and 5 are visible. In Position IV, faces 2, 4 and 6 are visible. Which number is opposite to 1?

4
5
2
6

Explanation: From Position I, 1 is adjacent to 2 and 3. From Position II, 1 is adjacent to 2 and 4. From Position III, 1 is adjacent to 3 and 5. Thus 1 cannot be opposite 2, 3, 4, or 5. The only remaining face is 6, which must be opposite to 1.

A cuboid of dimensions 4 cm × 5 cm × 6 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly one painted face?

36
60
44
52

Explanation: For a cuboid, cubes with exactly one painted face lie at the center of each face. The formula is: 2[(a − 2)(b − 2) + (b − 2)(c − 2) + (c − 2)(a − 2)]. Substituting: 2[(2 × 3) + (3 × 4) + (4 × 2)] = 2 × 26 = 52.

A 4 cm cube is painted on exactly three faces that meet at a common corner, and then cut into 1 cm³ cubes. How many smaller cubes have paint on at least one face?

52
37
48
28

Explanation: Using the principle of inclusion-exclusion for the three painted faces: each face has 4² = 16 painted cubes. The three pairs of adjacent painted faces share 3 edges with 4 cubes each. The common corner is counted three times and must be added back once. Thus: 3 × 16 − 3 × 4 + 1 = 48 − 12 + 1 = 37.

A cuboid of dimensions 3 cm × 4 cm × 5 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces?

16
20
24
28

Explanation: For a cuboid, cubes with exactly two painted faces lie along the edges excluding corners. The formula is: 4[(a − 2) + (b − 2) + (c − 2)]. Substituting: 4[(3 − 2) + (4 − 2) + (5 − 2)] = 4 × 6 = 24.

A 7 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and lie on the edges of the top face only?

28
35
20
14

Explanation: The top face has 4 edges. Each edge contains (n − 2) non-corner cubes with exactly two painted faces (painted on the top face and one adjacent side face), excluding the corner cubes which have three painted faces. Thus: 4 × (7 − 2) = 4 × 5 = 20.

A 9 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and are located on the four vertical edges only?

42
28
36
14

Explanation: A cube has 4 vertical edges. Each vertical edge contains (n − 2) cubes with exactly two painted faces (painted on two adjacent vertical side faces), excluding the top and bottom corner cubes. Thus: 4 × (9 − 2) = 4 × 7 = 28.

On a standard die, what is the sum of the numbers on the four faces adjacent to the face showing 4?

14
13
10
15

Explanation: On a standard die, opposite faces sum to 7, so 4 is opposite 3. The four faces adjacent to 4 are therefore 1, 2, 5, and 6. Their sum is 1 + 2 + 5 + 6 = 14.

A 3 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have no paint on any face?

0
6
8
1

Explanation: The unpainted interior forms a cube of side (n − 2). For n = 3, this gives (3 − 2)³ = 1³ = 1 cube at the very center.

A cube of side 6 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face only?

64
72
84
96

Explanation: Cubes with exactly one painted face lie at the center of each outer face, forming a square of side (n − 2). With 6 faces, the formula is: 6(n − 2)² = 6 × 16 = 96.

On a die, the numbers 1, 2 and 3 appear on three faces meeting at a corner and are arranged clockwise around that corner. If the die is placed with 2 on the top face and 3 on the front face, which number is on the right face?

5
6
1
4

Explanation: Since 1, 2 and 3 are arranged clockwise around their common corner, if 2 is on top and 3 is on the front, then moving clockwise around that corner from above places 1 on the right face. This follows directly from the given clockwise order of the three numbers around the corner.

A 5 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have either exactly three painted faces or exactly one painted face?

46
54
72
62

Explanation: Cubes with exactly three painted faces are always the 8 corner cubes. Cubes with exactly one painted face total 6(n − 2)² = 6 × 9 = 54. Adding these two groups: 8 + 54 = 62.

A 4 cm cube is painted on exactly two opposite faces only and then cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face?

48
32
24
16

Explanation: When only two opposite faces are painted, every unit cube on those two faces has exactly one painted face because all adjacent faces are unpainted. Each painted face contains n² = 16 cubes, so the total is 2 × 16 = 32.

An 8 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and lie on the edges of the top face only?

24
16
48
32

Explanation: The top face has 4 edges. Each edge contains (n − 2) cubes with exactly two painted faces (painted on the top face and one adjacent side face), excluding the corner cubes which have three painted faces. Thus: 4 × (8 − 2) = 4 × 6 = 24.

Three positions of the same die are shown. In Position I, faces 2, 3 and 4 are visible. In Position II, faces 2, 5 and 6 are visible. In Position III, faces 1, 3 and 5 are visible. Which number is opposite to 5?

2
4
6
1

Explanation: From Position II, 5 is adjacent to 2 and 6, so it cannot be opposite either. From Position III, 5 is adjacent to 1 and 3, so it cannot be opposite either. The only remaining face is 4, which must be opposite to 5.

A 6 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face and lie on the four vertical side faces only?

56
48
72
64

Explanation: There are 4 vertical side faces. On each face, the cubes with exactly one painted face form a square of side (n − 2) at the center. Therefore: 4 × (6 − 2)² = 4 × 16 = 64.

Three different positions of the same die are shown. In Position I, faces 1, 2 and 3 are visible. In Position II, faces 1, 4 and 5 are visible. In Position III, faces 2, 4 and 6 are visible. Which number is opposite to 1?

6
5
4
2

Explanation: From Position I, 1 is adjacent to 2 and 3, so it cannot be opposite either of them. From Position II, 1 is adjacent to 4 and 5, so it cannot be opposite either of them. The only remaining face is 6, which must be opposite to 1.

A 5 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face and lie on the four vertical side faces only (excluding the top and bottom faces)?

36
48
54
24

Explanation: There are 4 vertical side faces. On each face, the cubes with exactly one painted face form a square of side (n − 2) at the center, giving (5 − 2)² = 9 cubes per face. Therefore: 4 × 9 = 36.

A standard die is placed with 1 on the top face. It is rolled once toward the south (front face comes up), then once toward the east (right face comes up), and then once more toward the south. What number is on the top face after these three rolls?

6
4
1
5

Explanation: On a standard die with 1 opposite 6, 2 opposite 5, and 3 opposite 4, track the orientation through each roll. Starting with 1 on top: rolling south brings the front face (2) to the top; rolling east brings the right face (3) to the top; rolling south again brings the front face (6) to the top. The final top face is 6.

On a standard die with 1 opposite 6, 2 opposite 5, and 3 opposite 4, the die is placed with 1 on top and 2 on the right face. What is the sum of the numbers on the top, bottom, front, and back faces?

12
13
14
16

Explanation: With 1 on top, the bottom is 6. With 2 on the right, the left is 5. The remaining faces 3 and 4 must occupy the front and back positions. The sum is 1 + 6 + 3 + 4 = 14.

A 3 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces?

12
10
16
8

Explanation: For an n×n×n cube, the cubes with exactly two painted faces lie along the 12 edges, with each edge contributing (n − 2) cubes. Thus: 12 × (3 − 2) = 12 × 1 = 12.

What is the sum of the numbers on all six faces of a standard die?

28
18
21
24

Explanation: A standard die has faces numbered 1 through 6. The sum is 1 + 2 + 3 + 4 + 5 + 6 = 21.

A 7 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and are NOT located at any corner?

72
60
52
48

Explanation: By definition, cubes with exactly two painted faces lie strictly along the edges between the corners. There are 12 edges, and each edge contributes (n − 2) such cubes. Thus: 12 × (7 − 2) = 12 × 5 = 60.

A 6 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and lie on the edges of the top and bottom faces combined?

40
32
24
28

Explanation: The top face has 4 edges and the bottom face has 4 edges. Each edge contains (n − 2) non-corner cubes with exactly two painted faces (one face being the top or bottom, and the other being an adjacent side face). Thus: 8 × (6 − 2) = 8 × 4 = 32.

On a standard die, 4 is on the bottom face and 2 is on the front face. Which number is on the top face?

6
5
3
1

Explanation: On a standard die, opposite faces always sum to 7. Since 4 is on the bottom, the top face must be its opposite, which is 3 (because 4 + 3 = 7). The position of the front face does not alter the opposite-face relationship.

A 4 cm cube is painted on exactly two adjacent faces only and then cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face?

24
16
28
20

Explanation: When two adjacent faces are painted, the n cubes along their common edge have two painted faces. On each painted face, the remaining n² − n cubes have exactly one painted face. Thus: 2 × (4² − 4) = 2 × 12 = 24.

A 9 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face?

336
264
294
216

Explanation: Cubes with exactly one painted face lie at the center of each face, forming a square of side (n − 2). With 6 faces, the formula is: 6(n − 2)² = 6 × 7² = 6 × 49 = 294.

A 6 cm cube is painted on all faces and cut into 1 cm³ cubes. If all cubes having exactly one painted face are removed, how many cubes remain?

144
108
96
120

Explanation: The total number of unit cubes is 6³ = 216. The cubes with exactly one painted face total 6(n − 2)² = 6 × 16 = 96. Removing these: 216 − 96 = 120 cubes remain.

A 2 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces?

4
2
8
0

Explanation: A 2 cm cube cut into 1 cm³ cubes yields only 8 unit cubes, all of which are corner cubes. Every corner cube has exactly three painted faces, so there are no cubes with exactly two painted faces.

A cube is painted such that two opposite faces are red, two opposite faces are green, and the remaining two opposite faces are blue. It is then cut into 3×3×3 smaller cubes. How many smaller cubes have three different colors on them?

12
0
8
6

Explanation: Only the corner cubes have three faces exposed. At each corner, the three exposed faces belong to three different pairs of opposite faces, so each corner cube shows one red, one green, and one blue face. Since a cube has 8 corners, there are 8 such cubes.

A cuboid of dimensions 5 cm × 5 cm × 7 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have no paint on any face?

30
27
45
75

Explanation: For a cuboid, the unpainted interior cubes form a smaller cuboid of dimensions (a − 2) × (b − 2) × (c − 2). Substituting: (5 − 2) × (5 − 2) × (7 − 2) = 3 × 3 × 5 = 45.

A 4 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and are located on the edges of the bottom face only?

16
8
4
12

Explanation: The bottom face has 4 edges. Each edge contains (n − 2) cubes with exactly two painted faces (painted on the bottom face and one adjacent side face), excluding the corner cubes. Therefore: 4 × (4 − 2) = 4 × 2 = 8.

A standard die is placed with 3 on the top face. It is rolled once to the east (right), and then once to the north (away from you). What number appears on the top face after these two rolls?

6
5
1
4

Explanation: On a standard die with 1 opposite 6, 2 opposite 5, and 3 opposite 4, the faces adjacent to 3 are 1, 2, 5, and 6. Rolling east brings the right face to the top. Rolling north then brings the back face to the top. Tracking the orientation through both rolls: after the first roll the top becomes 2, and after the second roll the top becomes 6.

A standard die is placed with 6 on the top face and 3 on the front face. What is the sum of the numbers on the bottom, left, and right faces?

7
8
6
9

Explanation: With 6 on the top face, the bottom face is 1 (opposite faces sum to 7). With 3 on the front face, the back face is 4. The remaining faces 2 and 5 occupy the left and right positions. The sum of the bottom, left, and right faces is 1 + 2 + 5 = 8, regardless of which of 2 or 5 is on the left or right.

A cuboid of dimensions 5 cm × 7 cm × 9 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have no paint on any face?

105
189
63
135

Explanation: For a cuboid of dimensions a × b × c, the unpainted interior cubes form a smaller cuboid of dimensions (a − 2) × (b − 2) × (c − 2). Substituting the values: (5 − 2) × (7 − 2) × (9 − 2) = 3 × 5 × 7 = 105.

A cube of side 7 cm is painted on all faces and cut into 1 cm³ cubes. What is the total number of cubes that have either exactly three painted faces or no painted faces at all?

141
117
125
133

Explanation: Cubes with exactly three painted faces are always the 8 corner cubes. Cubes with no paint form the inner core of dimensions (n − 2)³. Adding these: 8 + (7 − 2)³ = 8 + 125 = 133.

A cuboid of dimensions 5 cm × 6 cm × 7 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly one painted face and lie on the largest face only?

12
18
20
30

Explanation: The largest face measures 6 cm × 7 cm. The cubes with exactly one painted face on this face form a rectangle at its center of dimensions (6 − 2) × (7 − 2). Thus: 4 × 5 = 20.

Four positions of the same die are shown. In Position I, faces 1, 2 and 3 are visible. In Position II, faces 1, 4 and 5 are visible. In Position III, faces 2, 4 and 6 are visible. In Position IV, faces 3, 5 and 6 are visible. If the die is placed with 3 on the top face, which number is on the bottom face?

1
4
2
6

Explanation: From Positions I and II, 1 is adjacent to 2, 3, 4, and 5, so 1 must be opposite 6. From Positions I and III, 2 is adjacent to 1, 3, 4, and 6, so 2 must be opposite 5. The remaining pair is 3 opposite 4. Therefore, with 3 on top, the bottom face shows 4.

A 4 cm cube is painted on all faces and sliced into 1 cm³ cubes. How many smaller cubes have paint on at least two faces?

32
24
16
40

Explanation: Cubes with at least two painted faces include the 8 corner cubes (three painted faces) and the cubes lying strictly along the edges (two painted faces). Using the formulas: 12(n − 2) + 8 = 12 × 2 + 8 = 32.

A cuboid of dimensions 6 cm × 8 cm × 10 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have no paint on any face?

240
192
144
216

Explanation: For a cuboid, the unpainted interior cubes form a smaller cuboid of dimensions (a − 2) × (b − 2) × (c − 2). Substituting: (6 − 2) × (8 − 2) × (10 − 2) = 4 × 6 × 8 = 192.

On a standard die, 1 is on the top face and 2 is on the front face. What is the product of the numbers on the four lateral (side) faces?

120
96
144
108

Explanation: With 1 on top, the bottom is 6. With 2 on the front, the back is 5. The remaining faces 3 and 4 occupy the right and left positions. The product of the four lateral faces is 2 × 5 × 3 × 4 = 120.

A 5 cm cube is painted on exactly four faces, leaving the top and bottom faces unpainted. It is then cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face?

60
40
72
48

Explanation: When the four vertical faces are painted, each painted face has n² = 25 unit cubes with paint. The two vertical edges on each painted face contain n = 5 cubes each that have two painted faces (shared with adjacent painted faces). Thus, on each face, the cubes with exactly one painted face number n² − 2n = 25 − 10 = 15. For four faces: 4 × 15 = 60.

A 10 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly two painted faces and are NOT located on any edge of the top or bottom face?

32
64
48
24

Explanation: The cubes with exactly two painted faces that are not on the top or bottom edges must lie on the 4 vertical edges. Each vertical edge contributes (n − 2) such cubes. Thus: 4 × (10 − 2) = 4 × 8 = 32.

Three positions of the same die are shown. In Position I, faces 1, 2 and 3 meet at a corner. In Position II, faces 1, 4 and 5 meet at a corner. In Position III, faces 3, 4 and 6 meet at a corner. Which number is opposite to 4?

5
3
2
1

Explanation: From Position II, 4 is adjacent to 1 and 5, so it cannot be opposite either. From Position III, 4 is adjacent to 3 and 6, so it cannot be opposite either. The only remaining face is 2, which must be opposite to 4.

A 6 cm cube is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have paint on exactly one face and lie on the top face only?

4
16
24
12

Explanation: On any single face of an n×n×n cube, the cubes with exactly one painted face form a central square of side (n − 2) at the center of that face. For the top face, this is a 4 × 4 square: (6 − 2)² = 4² = 16.

A cuboid of dimensions 4 cm × 5 cm × 6 cm is painted on all faces and cut into 1 cm³ cubes. How many smaller cubes have exactly three painted faces?

4
12
6
8

Explanation: In any cuboid, exactly three faces are exposed only at the 8 corner cubes, regardless of the specific dimensions. Therefore, there are always 8 such cubes.

A cube of side n cm is painted on all faces and cut into 1 cm³ cubes. If exactly 150 smaller cubes have paint on only one face, what is the value of n?

6
5
7
8

Explanation: Using the formula for cubes with exactly one painted face: 6(n − 2)² = 150. Dividing by 6 gives (n − 2)² = 25. Taking the square root: n − 2 = 5, so n = 7.

A cube of side n cm is painted on all faces and cut into 1 cm³ cubes. If exactly 60 smaller cubes have paint on exactly two faces, what is the value of n?

5
6
8
7

Explanation: Using the formula for cubes with exactly two painted faces: 12(n − 2) = 60. Dividing by 12 gives n − 2 = 5, so n = 7.

On a standard die with 1 opposite 6, 2 opposite 5, and 3 opposite 4, the die is placed with 4 on the bottom face and 5 on the back face. What is the sum of the numbers on the top and front faces?

4
6
5
7

Explanation: Since 4 is on the bottom, the top face is its opposite, which is 3. Since 5 is on the back, the front face is its opposite, which is 2. The sum is 3 + 2 = 5.

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