Cubes and Dice Practice Test Questions - Non Verbal Reasoning

The top face (2) is common to both positions.
In Position 1, Right = 1 and Front = 3, so the hidden Left face is unknown from this view alone.
In Position 2 (the die turned, top still 2), Front = 6 -- this is the value that was on the hidden Left face in Position 1.
Right and Left are always opposite faces on a cube, so the face opposite 1 is 6, which means the face opposite 6 is 1.

Right(1) & Left are always opposite
Position 2 Front(6) = Position 1's hidden Left
∴ Left = 6, so opposite of 1 is 6
∴ Opposite of 6 = 1 ✓

In this net, the four squares attached directly to the centre square (#, %, &, *) fold up to become the four side faces running around the centre.
The square that extends one step further out beyond one of those side squares (here, $, attached beyond #) always folds around to land on the face directly behind the centre -- that is, opposite to it.
So the face opposite @ is $.

Centre & the far extension square are always opposite
@ is centre, $ extends beyond #
∴ Opposite of @ = $ ✓

An ordinary (standard) dice always has its opposite faces adding up to 7 -- this is a fixed property, unlike the two- and three-view puzzles where the number arrangement is custom and must be worked out from the given views.
Here the face shown is 4, so the face opposite it is 7 - 4 = 3.

Standard dice: opposite faces always sum to 7
Opposite of 4 = 7 - 4
∴ Opposite of 4 = 3 ✓

The top face (Blue) is common to both positions.
In Position 1, Right = Yellow and Front = Red, so the hidden Left face is unknown from this view alone.
In Position 2 (top still Blue), Front = Black -- this is the colour that was on the hidden Left face in Position 1.
Right and Left are always opposite faces on a cube, so the face opposite Yellow is Black, which means the face opposite Black is Yellow.

Right(Yellow) & Left are always opposite
Position 2 Front(Black) = Position 1's hidden Left
∴ Opposite of Black = Yellow ✓

When a cube is cut into just 2 equal parts along each edge (2 × 2 × 2 = 8 smaller cubes), every single piece is a corner piece -- there is no room left for an edge, face-centre, or interior piece, since those categories only appear once the cube is cut into 3 or more parts per edge.
A corner piece always touches 3 outer faces, so all 8 of the smaller cubes have exactly 3 painted faces.

n = 2 → no edge/face-centre/interior pieces exist, only corners
Corners = 8 (all of them)
∴ Cubes with 3 painted faces = 8 ✓

The square that extends one step further out beyond one of the side arms (here, 5, attached beyond 4) always folds around to land on the face directly behind the centre square -- that is, opposite to it, no matter which arm carries the extension.
So the face opposite the centre (6) is 5.

Centre & the far extension square are always opposite
Centre = 6, extension = 5
∴ Opposite of 6 = 5 ✓

The top face (Q) is common to all three positions.
Position 1 shows Front = R, Right = P.
Turning the die 90 degrees (top still Q) brings the hidden Left face to the front: Position 2 shows Front = U, so Position 1's hidden Left face was U.
Turning another 90 degrees -- a full 180-degree turn from Position 1 -- brings the hidden Back face to the front: Position 3 shows Front = S, so Position 1's hidden Back face was S.
Front and Back are always opposite faces, so the face opposite R is S.

Front & Back are always opposite
Position 1 Front = R
Position 3 is a 180° turn, so its Front(S) = Position 1's hidden Back
∴ Opposite of R = S ✓

The centre square (5) has four squares attached directly to its four sides: 2 (above), 9 (below), 7 (right) and 3 (left), plus one more square (8) extending further out beyond 3.
When folded, the two squares attached directly above and below the centre (2 and 9) become opposite faces of the cube, and the two squares attached directly left and right of the centre (3 and 7) also become opposite faces of the cube -- regardless of where the extra extension square is.
So the face opposite 7 is 3.

Arms on opposite sides of the centre are opposite faces
East = 7, West = 3 (the extension 8 beyond West does not affect this pair)
∴ Opposite of 7 = 3 ✓

The four squares attached directly to the centre square (!) -- ?, ~, ^ and + -- fold up to become the four side faces running around the centre. The squares attached directly above and below the centre on the same arm (? and ~) always fold to become opposite faces of the cube, whichever arm carries the extra extension square.
So the face opposite ? is ~.

Arms directly above & below the centre are opposite faces
North = ?, South = ~
∴ Opposite of ? = ~ ✓

A face is adjacent to every other face except its own opposite, so "not adjacent to 2" simply means "opposite to 2".
The top face (1) is common to both positions.
In Position 1, Right = 2 and Front = 3, so the hidden Left face is unknown from this view alone.
In Position 2 (top still 1), Front = 5 -- this is the value that was on the hidden Left face in Position 1.
Right and Left are always opposite faces on a cube, so the face opposite 2 is 5 -- and that is the one face NOT adjacent to 2.

"Not adjacent to X" = "opposite of X"
Right(2) & Left are always opposite
Position 2 Front(5) = Position 1's hidden Left
∴ Face not adjacent to 2 = 5 ✓

The top face (2) is common to all three positions. Position 1 shows Front = 3, Right = 1.
Turning the die 90 degrees (top still 2) brings the hidden Left face to the front: Position 2 shows Front = 6, so Position 1's hidden Left face was 6.
Turning another 90 degrees -- a full 180-degree turn from Position 1 -- brings the hidden Back face to the front: Position 3 shows Front = 4, so Position 1's hidden Back face was 4.
Front and Back are always opposite faces, so the face opposite 3 is 4.

Front & Back are always opposite
Position 1 Front = 3
Position 3 is a 180° turn, so its Front(4) = Position 1's hidden Back
∴ Opposite of 3 = 4 ✓

The square that extends one step further out beyond one of the side arms (here, G, attached beyond J) always folds around to land on the face directly behind the centre square -- that is, opposite to it.
So the face opposite M is G.

Centre & the far extension square are always opposite
Centre = M, extension = G
∴ Opposite of M = G ✓

On a standard dice, the bottom face is always 7 minus the top face.
Die 1: bottom = 7 - 2 = 5.
Die 2: bottom = 7 - 5 = 2.
Die 3: bottom = 7 - 3 = 4.
Sum of the three bottom faces = 5 + 2 + 4 = 11.

Bottom = 7 - top, for each die
7-2=5, 7-5=2, 7-3=4
∴ Sum of bottoms = 5+2+4 = 11 ✓

The top face (E) is common to all three positions. Position 1 shows Front = F, Right = D.
Turning the die 90 degrees (top still E) brings the hidden Left face to the front: Position 2 shows Front = I, so Position 1's hidden Left face was I.
Turning another 90 degrees -- a full 180-degree turn from Position 1 -- brings the hidden Back face to the front: Position 3 shows Front = G, so Position 1's hidden Back face was G.
Front and Back are always opposite faces, so the face opposite F is G.

Front & Back are always opposite
Position 1 Front = F
Position 3 is a 180° turn, so its Front(G) = Position 1's hidden Back
∴ Opposite of F = G ✓

Cutting a painted cube into 4 equal parts along each edge (4 × 4 × 4 = 64 smaller cubes): the cubes with exactly 1 painted face are the ones sitting at the centre of each outer face, away from every edge.
On each face of the big cube, the face-centre cubes form a (4-2) × (4-2) = 2 × 2 = 4 square.
There are 6 faces, so the total is 6 × 4 = 24.

Face-centre cubes per face = (n-2)²
n = 4 → (4-2)² = 4 per face
6 faces × 4 = 24
∴ Cubes with exactly 1 painted face = 24 ✓

The centre square ({) has four squares attached directly to its sides: } (above), [ (below), ] (right) and ; (left), plus one more square (:) extending further out beyond ].
The two squares attached directly above and below the centre (} and [) always become opposite faces of the cube when folded, regardless of where the extension square is.
So the face opposite } is [.

Arms directly above & below the centre are opposite faces
North = }, South = [
∴ Opposite of } = [ ✓

For a box of size a × b × c painted on all outer faces and cut into unit cubes, the number of pieces with exactly 1 painted face is 2[(a-2)(b-2) + (b-2)(c-2) + (c-2)(a-2)].
Here a = 4, b = 3, c = 2, so (a-2) = 2, (b-2) = 1, (c-2) = 0.
2[(2 × 1) + (1 × 0) + (0 × 2)] = 2[2 + 0 + 0] = 4.
So the number of unit cubes with exactly 1 painted face is 4.

1-face count = 2[(a-2)(b-2)+(b-2)(c-2)+(c-2)(a-2)]
a=4,b=3,c=2 → (a-2,b-2,c-2)=(2,1,0)
2[(2×1)+(1×0)+(0×2)] = 2×2
∴ Cubes with exactly 1 painted face = 4 ✓

For a box of size a × b × c painted on all outer faces and cut into unit cubes, the number of pieces with exactly 2 painted faces (lying along an edge, but not at a corner) is 4[(a-2) + (b-2) + (c-2)].
Here a = 5, b = 4, c = 3, so (a-2) = 3, (b-2) = 2, (c-2) = 1.
4[3 + 2 + 1] = 4 × 6 = 24.
So the number of unit cubes with exactly 2 painted faces is 24.

2-face (edge) count = 4[(a-2)+(b-2)+(c-2)]
a=5,b=4,c=3 → (a-2,b-2,c-2)=(3,2,1)
4[3+2+1] = 4×6
∴ Cubes with exactly 2 painted faces = 24 ✓

Cutting a painted cube into 3 equal parts along each edge (3 × 3 × 3 = 27 smaller cubes) always produces 4 types of pieces:
the 8 corner cubes, each touching 3 outer faces, have 3 painted faces;
the cubes lying along an edge but not at a corner have exactly 2 painted faces -- there are 12 edges, and 1 such cube per edge (3 - 2 = 1), giving 12 cubes;
the cubes at the centre of each face have exactly 1 painted face -- 6 faces × 1 = 6 cubes;
the 1 cube right at the centre of the whole cube is not painted at all.
Check: 8 + 12 + 6 + 1 = 27.
So the number of smaller cubes with exactly 2 painted faces is 12.

For an n×n×n painted cube cut into n³ pieces:
corners (3 faces) = 8, edges (2 faces) = 12(n-2), face-centres (1 face) = 6(n-2)², core (0 faces) = (n-2)³
n = 3 → edges = 12 × 1 = 12
∴ Cubes with exactly 2 painted faces = 12 ✓

For an n × n × n cube painted on all outer faces and cut into n³ unit cubes, the number of pieces with exactly 2 painted faces (lying along an edge but not at a corner) is 12(n-2), since there are 12 edges and (n-2) such cubes along each edge.
Here n = 5, so 12(5-2) = 12 × 3 = 36.
So the number of unit cubes with exactly 2 painted faces is 36.

Edge cubes (2 faces) = 12(n-2)
n = 5 → 12 × 3
∴ Cubes with exactly 2 painted faces = 36 ✓

The three faces meeting at the corner diagonally opposite a given corner are simply the individual opposites of the three faces at the given corner -- reaching the far corner means crossing to the opposite face along all three directions at once.
Using the standard rule that opposite faces add up to 7: opposite of 1 is 6, opposite of 2 is 5, and opposite of 3 is 4.
Sum at the opposite corner = 6 + 5 + 4 = 15.

Opposite corner faces = 7 minus each given face
(7-1)+(7-2)+(7-3) = 6+5+4
∴ Sum at opposite corner = 15 ✓

Only the 4 side faces are painted here -- the top and bottom are left unpainted. A unit cube's painted-face count depends only on whether it lies on the outer boundary along the two painted horizontal directions; its position along the unpainted vertical direction never adds a painted face.
A unit cube has exactly 1 painted face when it sits on the boundary in exactly one of the two painted horizontal directions and in the middle position in the other direction. There are 4 such combinations (2 with the first direction on the boundary and the second in the middle, and 2 the other way round), and each combination can occur at any of the 3 positions along the unpainted vertical direction.
So the count is 4 × 3 = 12.

Only 4 side faces painted (top/bottom left plain)
Exactly-1-face combos (horizontal) = 4, free vertical positions = 3
4 × 3 = 12
∴ Cubes with exactly 1 painted face = 12 ✓

The solid is built from 3 stacked layers: the bottom layer is a 3 × 3 arrangement (9 cubes), the middle layer is a 2 × 2 arrangement (4 cubes), and the top layer is a single cube (1 cube).
Total number of cubes = 9 + 4 + 1 = 14.

Layer 1 = 3×3 = 9
Layer 2 = 2×2 = 4
Layer 3 = 1×1 = 1
∴ Total cubes = 9+4+1 = 14 ✓

The cube is cut into 4 × 4 × 4 = 64 smaller cubes. The unit cubes with 0 painted faces are the ones completely inside, away from every outer face -- they form a smaller (4-2) × (4-2) × (4-2) = 2 × 2 × 2 = 8 cube right at the centre.
Removing these 8 unpainted cubes leaves 64 - 8 = 56 smaller cubes.

Interior (0-face) cubes = (n-2)³
n = 4 → (4-2)³ = 8
Remaining = 64 - 8
∴ Cubes remaining = 56 ✓

The centre square (9) has four squares attached directly to its sides: 7 (above), 2 (below), 8 (right) and 3 (left), plus one more square (1) extending further out beyond 8.
The two squares attached directly above and below the centre (7 and 2) always become opposite faces of the cube when folded, regardless of where the extension square is.
So the face opposite 7 is 2.

Arms directly above & below the centre are opposite faces
North = 7, South = 2
∴ Opposite of 7 = 2 ✓

In the net, the centre square (filled circle) and the extension square (outline circle) are attached two steps apart along the same arm, so when folded they always land on opposite faces of the cube -- they can never be seen together on the same view.
Option 4 shows both the filled circle and the outline circle at once, so it cannot be formed by folding this sheet.
Options 1, 2 and 3 each show three faces that are mutually non-opposite (one face from each of the three opposite pairs), so all three are valid.

Centre & extension square are always opposite -- can never appear together
Option 4 shows both the filled circle (centre) & outline circle (extension)
∴ Option 4 cannot be formed ✓

In the net, the two arms attached directly above and below the centre (filled circle and outline circle) sit on opposite sides of the centre square, so when folded they always land on opposite faces of the cube.
Option 4 shows both the filled circle and the outline circle at once, so it cannot be formed by folding this sheet.
Options 1, 2 and 3 each show three mutually non-opposite faces, so all three are valid.

Arms directly above & below the centre are always opposite -- can never appear together
Option 4 shows both the filled circle & outline circle
∴ Option 4 cannot be formed ✓

In the net, the two arms attached directly left and right of the centre (filled circle and outline circle) sit on opposite sides of the centre square, so when folded they always land on opposite faces of the cube.
Option 4 shows both the filled circle and the outline circle at once, so it cannot be formed by folding this sheet.
Options 1, 2 and 3 each show three mutually non-opposite faces, so all three are valid.

Arms directly left & right of the centre are always opposite -- can never appear together
Option 4 shows both the filled circle & outline circle
∴ Option 4 cannot be formed ✓

In the net, the two arms attached directly above and below the centre (outline square and filled square) sit on opposite sides of the centre square, so when folded they always land on opposite faces of the cube.
Option 4 shows both the outline square and the filled square at once, so it cannot be formed by folding this sheet.
Options 1, 2 and 3 each show three mutually non-opposite faces, so all three are valid.

Arms directly above & below the centre are always opposite -- can never appear together
Option 4 shows both the outline square & filled square
∴ Option 4 cannot be formed ✓

The arms attached directly above and below the centre square (5 dots and 2 dots) sit on opposite sides of the centre, so when folded they always land on opposite faces of the cube, regardless of where the extension square is.
So the face opposite the 5-dot face has 2 dots.

Arms directly above & below the centre are opposite faces
North = 5 dots, South = 2 dots
∴ Opposite of 5 dots = 2 dots ✓

When a row of four squares is rolled into a tube to form four sides of a cube, each square keeps its own up/down edge pointing the same way throughout the fold. So the square attached above Q (K) always folds onto the top face, and the square attached below R (L) always folds onto the bottom face -- and top and bottom are always opposite each other.
So the face opposite K is L.

Tab above a row-square → top face; tab below a row-square → bottom face
Top & bottom are always opposite
∴ Opposite of K = L ✓

The three opposite pairs on this net are: filled/outline triangle (centre/extension), filled/outline circle (North/South arms), and filled/outline square (East/West arms).
Cube (1) shows a filled triangle, filled circle and filled square -- one face from each pair -- so it is valid.
Cube (2) shows both the filled and outline triangle together, which is impossible.
Cube (3) shows an outline triangle, outline circle and outline square -- one face from each pair -- so it is also valid.
Cube (4) shows both the filled and outline circle together, which is impossible.
So cubes (1) and (3) can be formed.

(1) and (3) = one face from each opposite pair ✓
(2) shows both triangles, (4) shows both circles → impossible
∴ Only (1) and (3) can be formed ✓

On this net, D and F are opposite (they are the 1st and 3rd squares of the row), E and G are opposite (2nd and 4th squares), and H and I are opposite (the tab above D and the tab below G).
Cube (1) shows D, E and H -- one face from each opposite pair -- so it is valid.
Cube (2) shows both D and F together, which is impossible.
Cube (3) shows F, G and I -- one face from each opposite pair -- so it is also valid.
Cube (4) shows both E and G together, which is impossible.
So cubes (1) and (3) can be formed.

Row squares 1 & 3 opposite, 2 & 4 opposite; top tab & bottom tab opposite
(1) and (3) = one face from each pair ✓
(2) shows D & F, (4) shows E & G → impossible
∴ Only (1) and (3) can be formed ✓

The square that extends one step further out beyond a side arm (here, the 3-dot square, attached beyond the 1-dot arm) always folds around to land on the face directly behind the centre square -- that is, opposite to it, no matter which arm carries the extension.
So the face opposite the centre (6 dots) has 3 dots.

Centre & the far extension square are always opposite
Centre = 6 dots, extension = 3 dots
∴ Opposite of 6 dots = 3 dots ✓

On this net, the 2-dot and 6-dot squares are opposite (1st and 3rd squares of the row), the 4-dot and 1-dot squares are opposite (2nd and 4th squares), and the 5-dot and 3-dot tabs are opposite each other.
So the face opposite the 4-dot face has 1 dot.

Row squares 1 & 3 opposite, 2 & 4 opposite
4-dot square is the 2nd square, 1-dot square is the 4th square
∴ Opposite of 4 dots = 1 dot ✓

Cubes (1), (3) and (4) each show one face from every opposite pair (outline/filled circle, outline/filled square, outline/filled triangle), so all three are valid.
Cube (2) shows both the outline and filled square together (the opposite North/South arms), which is impossible.
So cubes (1), (3) and (4) can be formed.

(1),(3),(4) = one face from each opposite pair ✓
(2) shows both squares together → impossible
∴ (1), (3) and (4) can be formed ✓

On this net, R and T are opposite (1st and 3rd squares), X and U are opposite (2nd and 4th squares), and V and W are opposite (the tab above U and the tab below R).
Cube (1) shows R, X and V -- one face from each opposite pair -- so it is valid.
Cube (2) shows both R and T together, which is impossible.
Cube (3) shows T, U and W -- one face from each opposite pair -- so it is also valid.
Cube (4) shows both X and U together, which is impossible.
So cubes (1) and (3) can be formed.

Row squares 1 & 3 opposite, 2 & 4 opposite; top tab & bottom tab opposite
(1) and (3) = one face from each pair ✓
(2) shows R & T, (4) shows X & U → impossible
∴ Only (1) and (3) can be formed ✓

The arms attached directly above and below the centre square (J and K) sit on opposite sides of the centre, so when folded they always land on opposite faces of the cube, regardless of where the extension square is.
So the face opposite J is K.

Arms directly above & below the centre are opposite faces
North = J, South = K
∴ Opposite of J = K ✓

On this net, the filled circle and the filled triangle are the 1st and 3rd squares of the row, so they are opposite; the filled square and the outline triangle are the 2nd and 4th squares, so they are opposite; and the outline circle and outline square tabs are opposite each other.
Cube (1) shows the filled circle and filled triangle together, which is impossible.
Cube (2) shows the filled circle, filled square and outline circle -- one face from each opposite pair -- so it is valid.
Cube (3) shows the filled triangle, outline triangle and outline square -- one face from each opposite pair -- so it is also valid.
Cube (4) shows the filled square and outline triangle together, which is impossible.
So cubes (2) and (3) can be formed.

Row squares 1 & 3 opposite, 2 & 4 opposite; top tab & bottom tab opposite
(2) and (3) = one face from each pair ✓
(1) shows both S1 & S3, (4) shows both S2 & S4 → impossible
∴ Only (2) and (3) can be formed ✓