Cubes and Dice

7 questions

All types7 Multiple choice3 Numeric3 Matching1

A standard six-sided die (where the sum of the numbers on any pair of opposite faces is exactly $7$ ) is placed on a flat table. In its initial position, the number $4$ is on the top face, the number $2$ is on the front face (facing you), and the number $1$ is on the right face.

The die is then rolled $90^\circ$ over its edges along the table in the following sequence:
1. Rolled to the Right (East).
2. Rolled Forward (North, away from you).
3. Rolled to the Right (East).
4. Rolled Backward (South, towards you).

What number will be on the top face of the die in its final position?

Show answer & explanation

Answer: 6

Answer

The number 6 will be on the top face.

By carefully tracking the 3D rotation of the die step-by-step, the top face transitions from 4 to 6 (after the first right roll), to 2 (after the forward roll), to 3 (after the second right roll), and finally to 6 (after the backward roll). The sequence of orthogonal rotations guarantees that the number 6 rests on the top face at the end.

Step-by-Step Solution

Determine the initial state of all six faces based on standard die rules.

Top = 4, Bottom = 3, Front = 2, Back = 5, Right = 1, Left = 6.

A standard die has opposite faces that sum to 7. This allows us to deduce the hidden faces from the visible ones.

Track face positions after the first roll: 90° to the Right (East).

Top becomes 6, Bottom becomes 1, Front remains 2, Back remains 5, Right becomes 4, Left becomes 3.

Rolling Right pivots the die over the right edge; the Left face (6) moves to the Top, while Front and Back remain unchanged.

Track face positions after the second roll: 90° Forward (North).

Top becomes 2, Bottom becomes 5, Front becomes 1, Back becomes 6, Right remains 4, Left remains 3.

Rolling Forward pivots over the back edge; the Front face (2) moves to the Top, while Right and Left remain unchanged.

Track face positions after the third roll: 90° to the Right (East).

Top becomes 3, Bottom becomes 4, Front remains 1, Back remains 6, Right becomes 2, Left becomes 5.

Rolling Right again moves the new Left face (3) to the Top, while Front and Back remain unchanged.

Track face positions after the final roll: 90° Backward (South).

Top becomes 6, Bottom becomes 1, Front becomes 3, Back becomes 4, Right remains 2, Left remains 5.

Rolling Backward pivots over the front edge; the Back face (6) moves to the Top. Thus, the final number on the top face is 6.

Key Concept

Spatial tracking of rolling dice and orientation of opposite faces.

Estimated Time:1m 30s

Question 2Question →

Three different positions of the same die are shown below. The six faces of the die are marked with the Greek letters $\alpha, \beta, \gamma, \delta, \epsilon$ , and $\zeta$ .

- Position 1: Top is $\alpha$ , Front is $\beta$ , Right is $\delta$
- Position 2: Top is $\beta$ , Front is $\gamma$ , Right is $\delta$
- Position 3: Top is $\beta$ , Front is $\alpha$ , Right is $\zeta$

Based on the given positions, which letter is on the face opposite to $\delta$ ?

\alpha

\beta

\gamma

\zeta

Show answer & explanation

Answer: $\zeta$

Answer

The letter on the face opposite to

\delta

\zeta

By analyzing the adjacent faces across the three positions, we can determine all opposite pairs. Gathering all unique adjacent faces for

\beta

, we find it is adjacent to

\alpha, \gamma, \delta

, and

\zeta

. This leaves

\epsilon

as the opposite face to

\beta

. Next, evaluating

\alpha

shows it is adjacent to

\beta, \delta, \zeta

, and

\epsilon

(since

\beta

and

\epsilon

are opposites), leaving

\gamma

as the opposite face to

\alpha

. By the process of elimination, the remaining two faces,

\delta

and

\zeta

, must be opposite to each other.

Step-by-Step Solution

Identify all visible adjacent faces for

\beta

across the given positions.

In Position 1,

\beta

is adjacent to

\alpha

and

\delta

. In Position 2,

\beta

is adjacent to

\gamma

and

\delta

. In Position 3,

\beta

is adjacent to

\alpha

and

\zeta

. Combining these,

\beta

is adjacent to

\alpha, \gamma, \delta

, and

\zeta

Since a single face of a die can have exactly four adjacent faces, finding all four uniquely determines the fifth face as its opposite.

Determine the face opposite to

\beta

Because

\alpha, \gamma, \delta

, and

\zeta

are adjacent to

\beta

, the only remaining letter,

\epsilon

, must be opposite to

\beta

By the process of elimination among the six faces, the one face not adjacent to a given face must lie on the opposite side.

Determine the faces adjacent and opposite to

\alpha

From Position 1 and 3,

\alpha

is adjacent to

\beta, \delta

, and

\zeta

. Because

\beta

is opposite

\epsilon

\alpha

must also be adjacent to

\epsilon

. This means

\alpha

is adjacent to

\beta, \delta, \epsilon

, and

\zeta

, leaving

\gamma

as the face opposite to

\alpha

Repeating the deduction process systematically for another face eliminates a second pair of opposite faces.

Identify the final opposite pair.

Since we have established that

\beta

is opposite

\epsilon

and

\alpha

is opposite

\gamma

, the remaining two faces,

\delta

and

\zeta

, must be opposite each other.

Once two pairs of opposite faces on a six-sided die are found, the last two remaining faces form the final opposite pair.

Key Concept

Identifying opposite faces on a 3D die by mapping adjacent faces from multiple rotational views.

Estimated Time:1m 30s

Question 3Question →

A large solid cube of side length $6 \text{ cm}$ has its opposite faces painted with the same color. The colors used are red, blue, and yellow. After the paint dries, the large cube is completely cut into smaller cubes of side length $1 \text{ cm}$ each. How many of these smaller cubes will have exactly two faces painted, with one face being red and the other face being yellow?

Show answer & explanation

Answer: 16

Answer

16 cubes

There are exactly 4 edges where a red face and a yellow face meet. Each of these edges contains exactly 4 middle cubes (excluding the corner cubes) that have exactly two faces painted. Therefore, multiplying

4 \text{ edges} \times 4 \text{ cubes}

gives 16.

Step-by-Step Solution

Calculate the number of smaller cubes along one edge of the large cube.

n = 6 / 1 = 6

cubes per edge.

The value

n

is required to determine how many cubes lie along any given dimension.

Determine the number of edges where a red face and a yellow face intersect.

There are exactly 4 intersecting edges.

The cube has 2 red faces (top/bottom) and 2 yellow faces (left/right). Each of the 2 red faces meets both of the 2 yellow faces, resulting in

2 \times 2 = 4

edges.

Calculate how many cubes on a single edge have exactly two faces painted.

n - 2 = 6 - 2 = 4

cubes per edge.

The cubes at the two extreme ends of any edge are corner cubes with three painted faces, so they must be subtracted from the total edge length.

Multiply the number of relevant edges by the number of two-face painted cubes per edge.

4 \text{ edges} \times 4 \text{ cubes} = 16

cubes.

This yields the total count of smaller cubes possessing exactly one red and one yellow painted face.

Key Concept

Calculation of painted smaller cubes resulting from cutting a colored larger cube based on intersecting faces.

Question 4Question →

A solid wooden block has dimensions $12 \text{ cm} \times 15 \text{ cm} \times 18 \text{ cm}$ . Three of its mutually adjacent faces (which meet at a single corner) are painted Red, and the remaining three faces are painted Blue. The block is then cut into smaller, identical cubes of size $1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm}$ . How many of these smaller cubes have AT LEAST one Red face AND at least one Blue face?

Show answer & explanation

Answer: 84

Answer

The unit cubes with both colors lie exactly on the 6 boundary edges separating the Red and Blue faces. By summing the lengths of these 6 edges (

12+12+15+15+18+18 = 90

) and subtracting the 6 overlapping corner cubes to correct for double-counting, we arrive at exactly 84 cubes.

Step-by-Step Solution

Identify the configuration of the painted faces on the block.

The 3 Red faces share one corner, and the 3 Blue faces share the diagonally opposite corner.

To accurately locate where the Red and Blue paints meet on the surface.

Determine which unit cubes satisfy the condition of having both Red and Blue faces.

Only the cubes lying exactly on the boundary edges between a Red face and a Blue face will have both colors.

Cubes fully within a Red face have no Blue paint, internal cubes have no paint, and purely corner cubes not on the boundary have only one color.

Count the number of boundary edges.

There are exactly 6 boundary edges forming a continuous zig-zag hexagonal ring around the block.

Each of the 3 Red faces shares an edge with exactly 2 Blue faces.

Determine the lengths of these 6 boundary edges.

The ring wraps around all three dimensions symmetrically, covering two edges of 12 cm, two of 15 cm, and two of 18 cm.

A cuboid has 4 edges of each dimension; the boundary ring traverses exactly half of them.

Calculate the total number of unit cubes on these edges using the inclusion-exclusion principle.

2 \times (12 + 15 + 18) - 6 = 90 - 6 = 84

cubes.

Summing the edge lengths gives 90 cubes, but the 6 corners connecting these edges in the ring are counted twice, so we must subtract 6 to prevent double-counting.

Key Concept

Spatial reasoning and painted cube boundary edge calculation.

Question 5Question →

A large solid cube of dimensions $5 \times 5 \times 5$ is painted using three colors: Red, Blue, and Green. The painting is applied such that the Top and Bottom faces are Red, the Front and Right faces are Blue, and the Back and Left faces are Green. The large cube is then cut into 125 smaller, identical cubes of dimensions $1 \times 1 \times 1$ . Match the categories of the smaller cubes described in List I with their exact counts in List II.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: Cubes with three different colors to 4; Cubes with two faces of the same color to 6; Cubes with one face Blue to 18; Cubes with no painted faces to 27.

The exact structural breakdown of the 5x5x5 cube under this specific adjacent-face coloring scheme yields exactly 4 three-color corners, 6 two-color edges (from 2 identical color intersections), 18 one-color blue face centers, and a 27-cube unpainted core.

Step-by-Step Solution

Analyze the corner cubes (exactly 3 faces painted) to match the first category.

There are 8 corners. Top-Front-Left is Red-Blue-Green (3 colors). Top-Front-Right is Red-Blue-Blue (2 colors). Analyzing all 8 corners yields exactly 4 corners that touch three different colors.

Because Front/Right are both Blue and Back/Left are both Green, corners sharing these adjacent identical faces will not have three different colors.

Analyze the edge cubes (exactly 2 faces painted) to match the second category.

There are 12 edges, each containing (5-2) = 3 cubes with exactly two painted faces. The condition requires both faces to be the same color. This only happens on the Front-Right edge (Blue-Blue) and Back-Left edge (Green-Green). Thus, 2 edges × 3 cubes = 6 cubes.

Edges represent the intersection of two faces. Only adjacent faces painted with the same color produce these specific cubes.

Analyze the face center cubes (exactly 1 face painted) to match the third category.

Each of the 6 faces has a central grid of (5-2)² = 9 cubes that have exactly one face painted. Since exactly 2 faces are painted Blue (Front and Right), there are 2 × 9 = 18 such cubes.

This singles out the purely Blue surface area minus edges and corners.

Calculate the completely unpainted inner core cubes to match the fourth category.

The unpainted inner volume is a smaller cube of dimensions (5-2) × (5-2) × (5-2) = 3³ = 27 cubes.

Removing the outer layer (1 cube thick) from all 6 sides leaves the unpainted core.

Key Concept

Advanced visualization of 3D spatial properties, edge intersections, and non-standard coloring rules on a subdivided cube.

Question 6Question →

A large solid cube is constructed using $125$ smaller, identical cubic blocks. Before the large cube is painted, all $8$ corner blocks are completely removed from the structure. The resulting modified structure is then completely dipped in a vat of red paint. After the paint dries, the structure is disassembled back into the remaining $117$ individual smaller blocks. How many of these $117$ smaller blocks have exactly two of their faces painted red?

Show answer & explanation

Answer: 12

Answer

In a standard

5 \times 5 \times 5

cube, the blocks with exactly two painted faces are located along the edges (excluding the corners). There are

5 - 2 = 3

such blocks per edge. However, when the

8

corner blocks are removed, the structural integrity changes. The edge blocks directly adjacent to the removed corners get an additional face exposed (the inner face that was previously touching the corner). This increases their painted faces from

2

3

. Only the single block in the exact middle of each of the

12

edges remains unaffected by the corner removal, keeping exactly

2

exposed faces. Thus, there are

12 \text{ edges} \times 1 \text{ block/edge} = 12

such blocks.

Step-by-Step Solution

Analyze the original edge composition of the 5x5x5 cube.

The cube has 12 edges. Each edge consists of 5 blocks: 2 corner blocks and 3 middle edge blocks.

To establish the baseline before the corners are removed.

Determine the spatial effect of removing the 8 corner blocks.

Removing a corner creates a 'dent'. The 3 edge blocks that were directly touching this corner now have an interior face exposed to the outside.

Paint will now reach surfaces that were previously hidden inside the structure.

Calculate the newly painted faces for the affected edge blocks.

The 2 edge blocks at the ends of each 3-block edge segment originally had 2 exposed faces. They now have 1 additional exposed face, bringing their total to 3 painted faces.

To eliminate these blocks from the target count of blocks with exactly 2 painted faces.

Identify the blocks that maintain exactly two exposed faces.

Only the single block in the exact center of each 5-block edge does not touch a corner. Its exposed faces remain unchanged at 2.

These are the only blocks in the entire modified structure that fit the criteria.

Calculate the final total.

12 edges multiplied by 1 block per edge equals 12 blocks.

To reach the final quantitative answer.

Key Concept

Spatial reasoning and dynamic visualization of modified 3D geometric structures.

Question 7Question →

A solid wooden cube is painted Black on two adjacent faces, while the remaining four faces are painted White. The large cube is then cut into $64$ identical smaller cubes. What is the total number of smaller cubes that have at least one Black face?

Show answer & explanation

Answer: 28

Answer

The large cube is divided into

64

smaller cubes, which means it forms a

4 \times 4 \times 4

grid. A single face of this large cube contains

4 \times 4 = 16

smaller cubes. Since two adjacent faces are painted Black, we count the cubes on both faces. The first face has

16

Black cubes. The second face also has

16

Black cubes, but it shares one edge with the first face. This shared edge consists of

4

smaller cubes. If we simply add

16 + 16 = 32

, we double-count the

4

cubes on the shared edge. Subtracting the overlap gives

32 - 4 = 28

unique smaller cubes that have at least one Black face.

Step-by-Step Solution

Find the edge length of the large cube in terms of smaller cubes.

n = 4

The total number of smaller cubes is

64

, and the total volume of a cube is

n^3

. Therefore,

n = \sqrt[3]{64} = 4

Count the smaller cubes on the first Black face.

16

cubes

Each face of an

n \times n \times n

cube contains

n^2

smaller cubes. Here,

4^2 = 16

cubes.

Count the smaller cubes on the second Black face and identify the overlap.

The second face adds

16

cubes, but the two faces share an edge of

4

cubes.

Since the two Black faces are adjacent, they intersect along one edge. The cubes on this edge are counted as part of both faces.

Calculate the total unique cubes with at least one Black face.

28

cubes

Using the inclusion-exclusion principle:

16

(first face) +

16

(second face) -

4

(shared edge) =

28

Key Concept

Calculation of painted smaller cubes from a larger cut cube, specifically handling adjacent face intersections.

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