← Four faces touch, exactly one is parallel · Net and Solid Structure

Four faces touch, exactly one is parallel · 12 practice problems

5.MD.C.36.G.A.4

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 medium answer: A

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked C\text{C}. (Ignore the orientation of the letters.)

F B C D A E D E C
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to C.

Givens
  • View 1 shows F on top, B front-left, C right.
  • View 2 shows D on top, A front-left, E right.
  • View 3 shows D on top, E front-left, C right.
Unknowns
  • The letter on the face parallel to C.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to C, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch C

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. C appears in View 1 with F and B, and in View 3 with D and E.
C touches: F, B (View 1),D, E (View 3)\text{C touches: } F,\ B\ (\text{View 1}),\quad D,\ E\ (\text{View 3})
Four different faces touch C: B, D, E, F.

2List all six faces and cross off C's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off C itself and the four faces that touch it (B, D, E, F). The only letter left over is A.
{A,B,C,D,E,F} ∖ {C, B,D,E,F}={A}\{A,B,C,D,E,F\}\ \setminus\ \{C,\ B,D,E,F\} = \{A\}
A is the only face never seen beside C.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to C (B, D, E, F), so the remaining face A must be the parallel one. That also fixes the other two pairs.
C∥A,B∥E,D∥FC \parallel A,\quad B \parallel E,\quad D \parallel F
Three pairs, and C is paired with A.
Answer: A
4 · Reviewdoes it hold up?

Check View 2, which does not show C at all: it shows D, A, E. A may appear there, and the other letters in it are neighbours of C -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is C, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 2 medium answer: A

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked D\text{D}. (Ignore the orientation of the letters.)

E B D C A F C F D
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to D.

Givens
  • View 1 shows E on top, B front-left, D right.
  • View 2 shows C on top, A front-left, F right.
  • View 3 shows C on top, F front-left, D right.
Unknowns
  • The letter on the face parallel to D.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to D, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch D

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. D appears in View 1 with E and B, and in View 3 with C and F.
D touches: E, B (View 1),C, F (View 3)\text{D touches: } E,\ B\ (\text{View 1}),\quad C,\ F\ (\text{View 3})
Four different faces touch D: B, C, E, F.

2List all six faces and cross off D's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off D itself and the four faces that touch it (B, C, E, F). The only letter left over is A.
{A,B,C,D,E,F} ∖ {D, B,C,E,F}={A}\{A,B,C,D,E,F\}\ \setminus\ \{D,\ B,C,E,F\} = \{A\}
A is the only face never seen beside D.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to D (B, C, E, F), so the remaining face A must be the parallel one. That also fixes the other two pairs.
D∥A,B∥F,C∥ED \parallel A,\quad B \parallel F,\quad C \parallel E
Three pairs, and D is paired with A.
Answer: A
4 · Reviewdoes it hold up?

Check View 2, which does not show D at all: it shows C, A, F. A may appear there, and the other letters in it are neighbours of D -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is D, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 3 medium answer: A

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked F\text{F}. (Ignore the orientation of the letters.)

E B F D A C D C F
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to F.

Givens
  • View 1 shows E on top, B front-left, F right.
  • View 2 shows D on top, A front-left, C right.
  • View 3 shows D on top, C front-left, F right.
Unknowns
  • The letter on the face parallel to F.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to F, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch F

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. F appears in View 1 with E and B, and in View 3 with D and C.
F touches: E, B (View 1),D, C (View 3)\text{F touches: } E,\ B\ (\text{View 1}),\quad D,\ C\ (\text{View 3})
Four different faces touch F: B, C, D, E.

2List all six faces and cross off F's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off F itself and the four faces that touch it (B, C, D, E). The only letter left over is A.
{A,B,C,D,E,F} ∖ {F, B,C,D,E}={A}\{A,B,C,D,E,F\}\ \setminus\ \{F,\ B,C,D,E\} = \{A\}
A is the only face never seen beside F.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to F (B, C, D, E), so the remaining face A must be the parallel one. That also fixes the other two pairs.
F∥A,B∥C,D∥EF \parallel A,\quad B \parallel C,\quad D \parallel E
Three pairs, and F is paired with A.
Answer: A
4 · Reviewdoes it hold up?

Check View 2, which does not show F at all: it shows D, A, C. A may appear there, and the other letters in it are neighbours of F -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is F, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 4 medium answer: D

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked E\text{E}. (Ignore the orientation of the letters.)

E B C B D C F A E
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to E.

Givens
  • View 1 shows E on top, B front-left, C right.
  • View 2 shows B on top, D front-left, C right.
  • View 3 shows F on top, A front-left, E right.
Unknowns
  • The letter on the face parallel to E.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to E, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch E

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. E appears in View 1 with B and C, and in View 3 with F and A.
E touches: B, C (View 1),F, A (View 3)\text{E touches: } B,\ C\ (\text{View 1}),\quad F,\ A\ (\text{View 3})
Four different faces touch E: A, B, C, F.

2List all six faces and cross off E's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off E itself and the four faces that touch it (A, B, C, F). The only letter left over is D.
{A,B,C,D,E,F} ∖ {E, A,B,C,F}={D}\{A,B,C,D,E,F\}\ \setminus\ \{E,\ A,B,C,F\} = \{D\}
D is the only face never seen beside E.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to E (A, B, C, F), so the remaining face D must be the parallel one. That also fixes the other two pairs.
E∥D,A∥C,B∥FE \parallel D,\quad A \parallel C,\quad B \parallel F
Three pairs, and E is paired with D.
Answer: D
4 · Reviewdoes it hold up?

Check View 2, which does not show E at all: it shows B, D, C. D may appear there, and the other letters in it are neighbours of E -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is E, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 5 medium answer: A

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked B\text{B}. (Ignore the orientation of the letters.)

F C B E A D E D B
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to B.

Givens
  • View 1 shows F on top, C front-left, B right.
  • View 2 shows E on top, A front-left, D right.
  • View 3 shows E on top, D front-left, B right.
Unknowns
  • The letter on the face parallel to B.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to B, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch B

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. B appears in View 1 with F and C, and in View 3 with E and D.
B touches: F, C (View 1),E, D (View 3)\text{B touches: } F,\ C\ (\text{View 1}),\quad E,\ D\ (\text{View 3})
Four different faces touch B: C, D, E, F.

2List all six faces and cross off B's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off B itself and the four faces that touch it (C, D, E, F). The only letter left over is A.
{A,B,C,D,E,F} ∖ {B, C,D,E,F}={A}\{A,B,C,D,E,F\}\ \setminus\ \{B,\ C,D,E,F\} = \{A\}
A is the only face never seen beside B.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to B (C, D, E, F), so the remaining face A must be the parallel one. That also fixes the other two pairs.
B∥A,C∥D,E∥FB \parallel A,\quad C \parallel D,\quad E \parallel F
Three pairs, and B is paired with A.
Answer: A
4 · Reviewdoes it hold up?

Check View 2, which does not show B at all: it shows E, A, D. A may appear there, and the other letters in it are neighbours of B -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is B, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 6 medium answer: D

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked C\text{C}. (Ignore the orientation of the letters.)

B C E D B E C A F
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to C.

Givens
  • View 1 shows B on top, C front-left, E right.
  • View 2 shows D on top, B front-left, E right.
  • View 3 shows C on top, A front-left, F right.
Unknowns
  • The letter on the face parallel to C.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to C, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch C

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. C appears in View 1 with B and E, and in View 3 with A and F.
C touches: B, E (View 1),A, F (View 3)\text{C touches: } B,\ E\ (\text{View 1}),\quad A,\ F\ (\text{View 3})
Four different faces touch C: A, B, E, F.

2List all six faces and cross off C's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off C itself and the four faces that touch it (A, B, E, F). The only letter left over is D.
{A,B,C,D,E,F} ∖ {C, A,B,E,F}={D}\{A,B,C,D,E,F\}\ \setminus\ \{C,\ A,B,E,F\} = \{D\}
D is the only face never seen beside C.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to C (A, B, E, F), so the remaining face D must be the parallel one. That also fixes the other two pairs.
C∥D,A∥E,B∥FC \parallel D,\quad A \parallel E,\quad B \parallel F
Three pairs, and C is paired with D.
Answer: D
4 · Reviewdoes it hold up?

Check View 2, which does not show C at all: it shows D, B, E. D may appear there, and the other letters in it are neighbours of C -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is C, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 7 medium answer: D

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked B\text{B}. (Ignore the orientation of the letters.)

F B C D F C B A E
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to B.

Givens
  • View 1 shows F on top, B front-left, C right.
  • View 2 shows D on top, F front-left, C right.
  • View 3 shows B on top, A front-left, E right.
Unknowns
  • The letter on the face parallel to B.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to B, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch B

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. B appears in View 1 with F and C, and in View 3 with A and E.
B touches: F, C (View 1),A, E (View 3)\text{B touches: } F,\ C\ (\text{View 1}),\quad A,\ E\ (\text{View 3})
Four different faces touch B: A, C, E, F.

2List all six faces and cross off B's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off B itself and the four faces that touch it (A, C, E, F). The only letter left over is D.
{A,B,C,D,E,F} ∖ {B, A,C,E,F}={D}\{A,B,C,D,E,F\}\ \setminus\ \{B,\ A,C,E,F\} = \{D\}
D is the only face never seen beside B.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to B (A, C, E, F), so the remaining face D must be the parallel one. That also fixes the other two pairs.
B∥D,A∥C,E∥FB \parallel D,\quad A \parallel C,\quad E \parallel F
Three pairs, and B is paired with D.
Answer: D
4 · Reviewdoes it hold up?

Check View 2, which does not show B at all: it shows D, F, C. D may appear there, and the other letters in it are neighbours of B -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is B, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 8 medium answer: B

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked A\text{A}. (Ignore the orientation of the letters.)

D A F E C B E A C
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to A.

Givens
  • View 1 shows D on top, A front-left, F right.
  • View 2 shows E on top, C front-left, B right.
  • View 3 shows E on top, A front-left, C right.
Unknowns
  • The letter on the face parallel to A.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to A, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch A

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. A appears in View 1 with D and F, and in View 3 with E and C.
A touches: D, F (View 1),E, C (View 3)\text{A touches: } D,\ F\ (\text{View 1}),\quad E,\ C\ (\text{View 3})
Four different faces touch A: C, D, E, F.

2List all six faces and cross off A's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off A itself and the four faces that touch it (C, D, E, F). The only letter left over is B.
{A,B,C,D,E,F} ∖ {A, C,D,E,F}={B}\{A,B,C,D,E,F\}\ \setminus\ \{A,\ C,D,E,F\} = \{B\}
B is the only face never seen beside A.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to A (C, D, E, F), so the remaining face B must be the parallel one. That also fixes the other two pairs.
A∥B,C∥F,D∥EA \parallel B,\quad C \parallel F,\quad D \parallel E
Three pairs, and A is paired with B.
Answer: B
4 · Reviewdoes it hold up?

Check View 2, which does not show A at all: it shows E, C, B. B may appear there, and the other letters in it are neighbours of A -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is A, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 9 medium answer: B

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked D\text{D}. (Ignore the orientation of the letters.)

D E F E B F C A D
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to D.

Givens
  • View 1 shows D on top, E front-left, F right.
  • View 2 shows E on top, B front-left, F right.
  • View 3 shows C on top, A front-left, D right.
Unknowns
  • The letter on the face parallel to D.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to D, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch D

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. D appears in View 1 with E and F, and in View 3 with C and A.
D touches: E, F (View 1),C, A (View 3)\text{D touches: } E,\ F\ (\text{View 1}),\quad C,\ A\ (\text{View 3})
Four different faces touch D: A, C, E, F.

2List all six faces and cross off D's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off D itself and the four faces that touch it (A, C, E, F). The only letter left over is B.
{A,B,C,D,E,F} ∖ {D, A,C,E,F}={B}\{A,B,C,D,E,F\}\ \setminus\ \{D,\ A,C,E,F\} = \{B\}
B is the only face never seen beside D.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to D (A, C, E, F), so the remaining face B must be the parallel one. That also fixes the other two pairs.
D∥B,A∥F,C∥ED \parallel B,\quad A \parallel F,\quad C \parallel E
Three pairs, and D is paired with B.
Answer: B
4 · Reviewdoes it hold up?

Check View 2, which does not show D at all: it shows E, B, F. B may appear there, and the other letters in it are neighbours of D -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is D, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 10 medium answer: A

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked E\text{E}. (Ignore the orientation of the letters.)

F B E C A D C D E
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to E.

Givens
  • View 1 shows F on top, B front-left, E right.
  • View 2 shows C on top, A front-left, D right.
  • View 3 shows C on top, D front-left, E right.
Unknowns
  • The letter on the face parallel to E.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to E, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch E

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. E appears in View 1 with F and B, and in View 3 with C and D.
E touches: F, B (View 1),C, D (View 3)\text{E touches: } F,\ B\ (\text{View 1}),\quad C,\ D\ (\text{View 3})
Four different faces touch E: B, C, D, F.

2List all six faces and cross off E's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off E itself and the four faces that touch it (B, C, D, F). The only letter left over is A.
{A,B,C,D,E,F} ∖ {E, B,C,D,F}={A}\{A,B,C,D,E,F\}\ \setminus\ \{E,\ B,C,D,F\} = \{A\}
A is the only face never seen beside E.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to E (B, C, D, F), so the remaining face A must be the parallel one. That also fixes the other two pairs.
E∥A,B∥D,C∥FE \parallel A,\quad B \parallel D,\quad C \parallel F
Three pairs, and E is paired with A.
Answer: A
4 · Reviewdoes it hold up?

Check View 2, which does not show E at all: it shows C, A, D. A may appear there, and the other letters in it are neighbours of E -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is E, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 11 medium answer: C

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked F\text{F}. (Ignore the orientation of the letters.)

F B D B C D E A F
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to F.

Givens
  • View 1 shows F on top, B front-left, D right.
  • View 2 shows B on top, C front-left, D right.
  • View 3 shows E on top, A front-left, F right.
Unknowns
  • The letter on the face parallel to F.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to F, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch F

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. F appears in View 1 with B and D, and in View 3 with E and A.
F touches: B, D (View 1),E, A (View 3)\text{F touches: } B,\ D\ (\text{View 1}),\quad E,\ A\ (\text{View 3})
Four different faces touch F: A, B, D, E.

2List all six faces and cross off F's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off F itself and the four faces that touch it (A, B, D, E). The only letter left over is C.
{A,B,C,D,E,F} ∖ {F, A,B,D,E}={C}\{A,B,C,D,E,F\}\ \setminus\ \{F,\ A,B,D,E\} = \{C\}
C is the only face never seen beside F.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to F (A, B, D, E), so the remaining face C must be the parallel one. That also fixes the other two pairs.
F∥C,A∥D,B∥EF \parallel C,\quad A \parallel D,\quad B \parallel E
Three pairs, and F is paired with C.
Answer: C
4 · Reviewdoes it hold up?

Check View 2, which does not show F at all: it shows B, C, D. C may appear there, and the other letters in it are neighbours of F -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is F, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.
Variant 12 medium answer: B

A cube has the letters A\text{A} through F\text{F} written on its six faces, and it is shown viewed from several directions. Find the letter on the face parallel to the face marked A\text{A}. (Ignore the orientation of the letters.)

D A E F C B F A C
Show solution
1 · Understandwhat's really being asked

One cube with letters A, B, C, D, E, F on its faces is drawn from three directions. We need the letter on the face parallel to A.

Givens
  • View 1 shows D on top, A front-left, E right.
  • View 2 shows F on top, C front-left, B right.
  • View 3 shows F on top, A front-left, C right.
Unknowns
  • The letter on the face parallel to A.
Constraints
  • A cube face touches four faces and is parallel to exactly one.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#10 Create a Physical Representation

Faces seen together at one corner must touch each other. Collect everything seen next to A, and the one letter never seen with it is the one on the far side.

3 · Execute3 carry out the plan

1Read which faces touch A

#17 Visualize Spatial Relationships 6.G.A.4
In any single corner view, the three labeled faces meet at one corner, so they are all next to each other. A appears in View 1 with D and E, and in View 3 with F and C.
A touches: D, E (View 1),F, C (View 3)\text{A touches: } D,\ E\ (\text{View 1}),\quad F,\ C\ (\text{View 3})
Four different faces touch A: C, D, E, F.

2List all six faces and cross off A's neighbours

#2 Make a Systematic List 5.MD.C.3
The cube uses the six letters A, B, C, D, E, F. Cross off A itself and the four faces that touch it (C, D, E, F). The only letter left over is B.
{A,B,C,D,E,F} ∖ {A, C,D,E,F}={B}\{A,B,C,D,E,F\}\ \setminus\ \{A,\ C,D,E,F\} = \{B\}
B is the only face never seen beside A.

3Confirm with the perpendicular/parallel rule

#10 Create a Physical Representation 6.G.A.4
A face is perpendicular to 4 faces and parallel to exactly 1. We found 4 faces perpendicular to A (C, D, E, F), so the remaining face B must be the parallel one. That also fixes the other two pairs.
A∥B,C∥E,D∥FA \parallel B,\quad C \parallel E,\quad D \parallel F
Three pairs, and A is paired with B.
Answer: B
4 · Reviewdoes it hold up?

Check View 2, which does not show A at all: it shows F, C, B. B may appear there, and the other letters in it are neighbours of A -- nothing contradicts the pairing.

Another way: Counting is enough on its own: six faces, one is A, four were seen touching it, so exactly one is left and it has to be the parallel face.

Standardsmin grade 6
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Knowing a cube has six faces in three opposite pairs.
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Reading corner views as statements about which faces touch.
💡Takeaway. Faces you can see together must touch, so the face you never see with it is the one on the opposite side.