← Paint depends on position: face, edge or corner · Cube Stacks and Their Views

Paint depends on position: face, edge or corner · 12 practice problems

6.G.A.46.G.A.2

Generated variants — 12

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

Variant 1 easy answer: 6 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 27 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 3 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 27 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 3 by 3; strip off the outer ring and 1 by 1 is left.
(3−2)×(3−2)=1(3 - 2) \times (3 - 2) = 1
1 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
1×6=61 \times 6 = 6
6 cubes with one painted face.
Answer: 6 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 12 + 6 + 1 = 27, which is 3 x 3 x 3.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 54 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 2 easy answer: 24 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 64 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 4 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 64 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 4 by 4; strip off the outer ring and 2 by 2 is left.
(4−2)×(4−2)=4(4 - 2) \times (4 - 2) = 4
4 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
4×6=244 \times 6 = 24
24 cubes with one painted face.
Answer: 24 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 24 + 24 + 8 = 64, which is 4 x 4 x 4.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 96 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 3 easy answer: 54 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 125 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 5 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 125 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 5 by 5; strip off the outer ring and 3 by 3 is left.
(5−2)×(5−2)=9(5 - 2) \times (5 - 2) = 9
9 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
9×6=549 \times 6 = 54
54 cubes with one painted face.
Answer: 54 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 36 + 54 + 27 = 125, which is 5 x 5 x 5.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 150 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 4 easy answer: 96 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 216 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 6 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 216 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 6 by 6; strip off the outer ring and 4 by 4 is left.
(6−2)×(6−2)=16(6 - 2) \times (6 - 2) = 16
16 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
16×6=9616 \times 6 = 96
96 cubes with one painted face.
Answer: 96 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 48 + 96 + 64 = 216, which is 6 x 6 x 6.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 216 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 5 medium answer: 150 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 343 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 7 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 343 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 7 by 7; strip off the outer ring and 5 by 5 is left.
(7−2)×(7−2)=25(7 - 2) \times (7 - 2) = 25
25 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
25×6=15025 \times 6 = 150
150 cubes with one painted face.
Answer: 150 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 60 + 150 + 125 = 343, which is 7 x 7 x 7.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 294 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 6 medium answer: 216 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 512 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 8 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 512 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 8 by 8; strip off the outer ring and 6 by 6 is left.
(8−2)×(8−2)=36(8 - 2) \times (8 - 2) = 36
36 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
36×6=21636 \times 6 = 216
216 cubes with one painted face.
Answer: 216 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 72 + 216 + 216 = 512, which is 8 x 8 x 8.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 384 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 7 medium answer: 294 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 729 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 9 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 729 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 9 by 9; strip off the outer ring and 7 by 7 is left.
(9−2)×(9−2)=49(9 - 2) \times (9 - 2) = 49
49 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
49×6=29449 \times 6 = 294
294 cubes with one painted face.
Answer: 294 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 84 + 294 + 343 = 729, which is 9 x 9 x 9.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 486 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 8 medium answer: 384 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 1000 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 10 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 1000 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 10 by 10; strip off the outer ring and 8 by 8 is left.
(10−2)×(10−2)=64(10 - 2) \times (10 - 2) = 64
64 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
64×6=38464 \times 6 = 384
384 cubes with one painted face.
Answer: 384 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 96 + 384 + 512 = 1000, which is 10 x 10 x 10.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 600 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 9 hard answer: 600 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 1728 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 12 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 1728 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 12 by 12; strip off the outer ring and 10 by 10 is left.
(12−2)×(12−2)=100(12 - 2) \times (12 - 2) = 100
100 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
100×6=600100 \times 6 = 600
600 cubes with one painted face.
Answer: 600 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 120 + 600 + 1000 = 1728, which is 12 x 12 x 12.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 864 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 10 hard answer: 1014 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 3375 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 15 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 3375 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 15 by 15; strip off the outer ring and 13 by 13 is left.
(15−2)×(15−2)=169(15 - 2) \times (15 - 2) = 169
169 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
169×6=1014169 \times 6 = 1014
1014 cubes with one painted face.
Answer: 1014 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 156 + 1014 + 2197 = 3375, which is 15 x 15 x 15.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 1350 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 11 hard answer: 1944 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 8000 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 20 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 8000 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 20 by 20; strip off the outer ring and 18 by 18 is left.
(20−2)×(20−2)=324(20 - 2) \times (20 - 2) = 324
324 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
324×6=1944324 \times 6 = 1944
1944 cubes with one painted face.
Answer: 1944 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 216 + 1944 + 5832 = 8000, which is 20 x 20 x 20.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 2400 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.
Variant 12 hard answer: 3174 cubes

Unit cubes are stacked into the cube shown at the right and the whole outside is painted. How many of the unit cubes have exactly one painted face? (The bottom face is painted too.)

Show solution
1 · Understandwhat's really being asked

A cube of 15625 unit cubes is painted all over, bottom included. We want how many have exactly one painted face.

Givens
  • The cube is 25 unit cubes along each edge.
  • Every outside surface is painted, including the bottom.
Unknowns
  • How many unit cubes have exactly one painted face.
Constraints
  • A cube's painted faces are exactly the faces that were on the outside.
2 · Planchoose the strategy

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

Do not count 15625 cubes one at a time. How much paint a cube gets depends only on where it sits -- corner, edge, face or inside -- so count positions instead.

3 · Execute4 carry out the plan

1Sort the cubes by where they sit

#17 Visualize Spatial Relationships 6.G.A.4
A corner cube shows three faces, an edge cube two, a cube in the middle of a face one, and a cube inside the solid none.
3, 2, 1, 03,\ 2,\ 1,\ 0
Four kinds of place, that is all.

2Find where one painted face happens

#10 Create a Physical Representation 6.G.A.4
Only strictly inside a face: not on an edge and not at a corner.
face centre→1\text{face centre} \rightarrow 1
Six faces to look at.

3Count one face's middle

#2 Make a Systematic List 6.G.A.2
A face is 25 by 25; strip off the outer ring and 23 by 23 is left.
(25−2)×(25−2)=529(25 - 2) \times (25 - 2) = 529
529 such cubes per face.

4Six faces, no overlaps

#2 Make a Systematic List 6.G.A.2
A cube strictly inside one face cannot be inside another, so nothing is counted twice.
529×6=3174529 \times 6 = 3174
3174 cubes with one painted face.
Answer: 3174 cubes
4 · Reviewdoes it hold up?

The four kinds account for the whole cube: 8 + 276 + 3174 + 12167 = 15625, which is 25 x 25 x 25.

Another way: Counting one face's middle and multiplying is the same as taking the whole surface, 3750 unit squares, and removing the edges and corners -- more arithmetic for the same figure.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Relating a small cube's painted faces to its position.
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in each kind of place.
💡Takeaway. How much paint a cube gets is decided by where it was standing. Count places, not cubes.