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Exposed faces depend on a cube's position · 12 practice problems

5.MD.C.35.MD.C.45.MD.C.5

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Generated variants — 12

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

Variant 1 easy

As shown at the right, $27$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly one painted face. (The faces touching the ground are also painted.)

The $27$ small cubes are packed $3$ across, $3$ deep, and $3$ high with no gaps, forming a large cube ( $3 \times 3 \times 3$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 27 small ones is painted all over, then pulled apart. We need how many small cubes show exactly one painted face.

Givens

The stack is 3 cubes across, 3 deep and 3 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly one painted face.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 27 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2One painted face pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly one painted side had exactly one of its sides facing out, and that happens at a face centre and nowhere else.

\text{face centre} \rightarrow 1

So we only need to look at the face centres.

3Count how many face centres a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 6 face centres.

6

6 face centres, always.

4Count the cubes in one face centre, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 3 cubes along a side, the middle of a face is that square with its border stripped off -- that is 1 small cube in one face centre. Multiply that by the 6 face centres.

(3 - 2) \times (3 - 2) = 1

6 x 1 = 6.

Answer: 6

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 1 + 6 x 1 + 1 = 27, which is 3 x 3 x 3 = 27. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 27 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 2 easy

As shown at the right, $27$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly two painted faces. (The faces touching the ground are also painted.)

The $27$ small cubes are packed $3$ across, $3$ deep, and $3$ high with no gaps, forming a large cube ( $3 \times 3 \times 3$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 27 small ones is painted all over, then pulled apart. We need how many small cubes show exactly two painted faces.

Givens

The stack is 3 cubes across, 3 deep and 3 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly two painted faces.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 27 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2Two painted faces pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly two painted sides had exactly two of its sides facing out, and that happens at an edge and nowhere else.

\text{edge} \rightarrow 2

So we only need to look at the edges.

3Count how many edges a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 12 edges.

12

12 edges, always.

4Count the cubes in one edge, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 3 cubes along a side, an edge runs between two corners, and those corners are already counted, so an edge holds the small cubes strictly between them -- that is 1 small cube in one edge. Multiply that by the 12 edges.

3 - 2 = 1

12 x 1 = 12.

Answer: 12

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 1 + 6 x 1 + 1 = 27, which is 3 x 3 x 3 = 27. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 27 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 3 easy

As shown at the right, $64$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly one painted face. (The faces touching the ground are also painted.)

The $64$ small cubes are packed $4$ across, $4$ deep, and $4$ high with no gaps, forming a large cube ( $4 \times 4 \times 4$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 64 small ones is painted all over, then pulled apart. We need how many small cubes show exactly one painted face.

Givens

The stack is 4 cubes across, 4 deep and 4 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly one painted face.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 64 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2One painted face pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly one painted side had exactly one of its sides facing out, and that happens at a face centre and nowhere else.

\text{face centre} \rightarrow 1

So we only need to look at the face centres.

3Count how many face centres a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 6 face centres.

6

6 face centres, always.

4Count the cubes in one face centre, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 4 cubes along a side, the middle of a face is that square with its border stripped off -- that is 4 small cubes in one face centre. Multiply that by the 6 face centres.

(4 - 2) \times (4 - 2) = 4

6 x 4 = 24.

Answer: 24

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 2 + 6 x 4 + 8 = 64, which is 4 x 4 x 4 = 64. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 64 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 4 medium

As shown at the right, $64$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly no painted faces. (The faces touching the ground are also painted.)

The $64$ small cubes are packed $4$ across, $4$ deep, and $4$ high with no gaps, forming a large cube ( $4 \times 4 \times 4$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 64 small ones is painted all over, then pulled apart. We need how many small cubes show exactly no painted faces.

Givens

The stack is 4 cubes across, 4 deep and 4 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly no painted faces.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 64 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2No painted faces pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly no painted sides had exactly no of its sides facing out, and that happens at the core and nowhere else.

\text{core} \rightarrow 0

So we only need to look at the cores.

3Count how many cores a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 1 core.

1

1 core, always.

4Count the cubes in one core, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 4 cubes along a side, the hidden core is the whole stack with one layer peeled off every side -- that is 8 small cubes in one core. There is only one core, so that is the whole count.

(4 - 2) \times (4 - 2) \times (4 - 2) = 8

1 x 8 = 8.

Answer: 8

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 2 + 6 x 4 + 8 = 64, which is 4 x 4 x 4 = 64. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 64 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 5 easy

As shown at the right, $64$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly two painted faces. (The faces touching the ground are also painted.)

The $64$ small cubes are packed $4$ across, $4$ deep, and $4$ high with no gaps, forming a large cube ( $4 \times 4 \times 4$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 64 small ones is painted all over, then pulled apart. We need how many small cubes show exactly two painted faces.

Givens

The stack is 4 cubes across, 4 deep and 4 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly two painted faces.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 64 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2Two painted faces pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly two painted sides had exactly two of its sides facing out, and that happens at an edge and nowhere else.

\text{edge} \rightarrow 2

So we only need to look at the edges.

3Count how many edges a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 12 edges.

12

12 edges, always.

4Count the cubes in one edge, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 4 cubes along a side, an edge runs between two corners, and those corners are already counted, so an edge holds the small cubes strictly between them -- that is 2 small cubes in one edge. Multiply that by the 12 edges.

4 - 2 = 2

12 x 2 = 24.

Answer: 24

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 2 + 6 x 4 + 8 = 64, which is 4 x 4 x 4 = 64. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 64 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 6 medium

As shown at the right, $125$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly two painted faces. (The faces touching the ground are also painted.)

The $125$ small cubes are packed $5$ across, $5$ deep, and $5$ high with no gaps, forming a large cube ( $5 \times 5 \times 5$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 125 small ones is painted all over, then pulled apart. We need how many small cubes show exactly two painted faces.

Givens

The stack is 5 cubes across, 5 deep and 5 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly two painted faces.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 125 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2Two painted faces pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly two painted sides had exactly two of its sides facing out, and that happens at an edge and nowhere else.

\text{edge} \rightarrow 2

So we only need to look at the edges.

3Count how many edges a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 12 edges.

12

12 edges, always.

4Count the cubes in one edge, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 5 cubes along a side, an edge runs between two corners, and those corners are already counted, so an edge holds the small cubes strictly between them -- that is 3 small cubes in one edge. Multiply that by the 12 edges.

5 - 2 = 3

12 x 3 = 36.

Answer: 36

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 3 + 6 x 9 + 27 = 125, which is 5 x 5 x 5 = 125. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 125 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 7 medium

As shown at the right, $125$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly one painted face. (The faces touching the ground are also painted.)

The $125$ small cubes are packed $5$ across, $5$ deep, and $5$ high with no gaps, forming a large cube ( $5 \times 5 \times 5$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 125 small ones is painted all over, then pulled apart. We need how many small cubes show exactly one painted face.

Givens

The stack is 5 cubes across, 5 deep and 5 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly one painted face.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 125 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2One painted face pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly one painted side had exactly one of its sides facing out, and that happens at a face centre and nowhere else.

\text{face centre} \rightarrow 1

So we only need to look at the face centres.

3Count how many face centres a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 6 face centres.

6

6 face centres, always.

4Count the cubes in one face centre, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 5 cubes along a side, the middle of a face is that square with its border stripped off -- that is 9 small cubes in one face centre. Multiply that by the 6 face centres.

(5 - 2) \times (5 - 2) = 9

6 x 9 = 54.

Answer: 54

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 3 + 6 x 9 + 27 = 125, which is 5 x 5 x 5 = 125. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 125 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 8 medium

As shown at the right, $125$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly no painted faces. (The faces touching the ground are also painted.)

The $125$ small cubes are packed $5$ across, $5$ deep, and $5$ high with no gaps, forming a large cube ( $5 \times 5 \times 5$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 125 small ones is painted all over, then pulled apart. We need how many small cubes show exactly no painted faces.

Givens

The stack is 5 cubes across, 5 deep and 5 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly no painted faces.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 125 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2No painted faces pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly no painted sides had exactly no of its sides facing out, and that happens at the core and nowhere else.

\text{core} \rightarrow 0

So we only need to look at the cores.

3Count how many cores a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 1 core.

1

1 core, always.

4Count the cubes in one core, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 5 cubes along a side, the hidden core is the whole stack with one layer peeled off every side -- that is 27 small cubes in one core. There is only one core, so that is the whole count.

(5 - 2) \times (5 - 2) \times (5 - 2) = 27

1 x 27 = 27.

Answer: 27

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 3 + 6 x 9 + 27 = 125, which is 5 x 5 x 5 = 125. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 125 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 9 hard

As shown at the right, $216$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly two painted faces. (The faces touching the ground are also painted.)

The $216$ small cubes are packed $6$ across, $6$ deep, and $6$ high with no gaps, forming a large cube ( $6 \times 6 \times 6$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 216 small ones is painted all over, then pulled apart. We need how many small cubes show exactly two painted faces.

Givens

The stack is 6 cubes across, 6 deep and 6 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly two painted faces.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 216 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2Two painted faces pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly two painted sides had exactly two of its sides facing out, and that happens at an edge and nowhere else.

\text{edge} \rightarrow 2

So we only need to look at the edges.

3Count how many edges a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 12 edges.

12

12 edges, always.

4Count the cubes in one edge, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 6 cubes along a side, an edge runs between two corners, and those corners are already counted, so an edge holds the small cubes strictly between them -- that is 4 small cubes in one edge. Multiply that by the 12 edges.

6 - 2 = 4

12 x 4 = 48.

Answer: 48

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 4 + 6 x 16 + 64 = 216, which is 6 x 6 x 6 = 216. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 216 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 10 hard

As shown at the right, $216$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly no painted faces. (The faces touching the ground are also painted.)

The $216$ small cubes are packed $6$ across, $6$ deep, and $6$ high with no gaps, forming a large cube ( $6 \times 6 \times 6$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 216 small ones is painted all over, then pulled apart. We need how many small cubes show exactly no painted faces.

Givens

The stack is 6 cubes across, 6 deep and 6 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly no painted faces.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 216 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2No painted faces pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly no painted sides had exactly no of its sides facing out, and that happens at the core and nowhere else.

\text{core} \rightarrow 0

So we only need to look at the cores.

3Count how many cores a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 1 core.

1

1 core, always.

4Count the cubes in one core, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 6 cubes along a side, the hidden core is the whole stack with one layer peeled off every side -- that is 64 small cubes in one core. There is only one core, so that is the whole count.

(6 - 2) \times (6 - 2) \times (6 - 2) = 64

1 x 64 = 64.

Answer: 64

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 4 + 6 x 16 + 64 = 216, which is 6 x 6 x 6 = 216. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 216 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 11 hard

As shown at the right, $216$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly one painted face. (The faces touching the ground are also painted.)

The $216$ small cubes are packed $6$ across, $6$ deep, and $6$ high with no gaps, forming a large cube ( $6 \times 6 \times 6$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 216 small ones is painted all over, then pulled apart. We need how many small cubes show exactly one painted face.

Givens

The stack is 6 cubes across, 6 deep and 6 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly one painted face.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 216 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2One painted face pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly one painted side had exactly one of its sides facing out, and that happens at a face centre and nowhere else.

\text{face centre} \rightarrow 1

So we only need to look at the face centres.

3Count how many face centres a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 6 face centres.

6

6 face centres, always.

4Count the cubes in one face centre, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 6 cubes along a side, the middle of a face is that square with its border stripped off -- that is 16 small cubes in one face centre. Multiply that by the 6 face centres.

(6 - 2) \times (6 - 2) = 16

6 x 16 = 96.

Answer: 96

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 4 + 6 x 16 + 64 = 216, which is 6 x 6 x 6 = 216. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 216 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Variant 12 hard

As shown at the right, $343$ small cubes are stacked into one large cube. The entire outer surface is painted, and then the small cubes are taken apart again. Find how many of the small cubes have exactly one painted face. (The faces touching the ground are also painted.)

The $343$ small cubes are packed $7$ across, $7$ deep, and $7$ high with no gaps, forming a large cube ( $7 \times 7 \times 7$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

A large cube built from 343 small ones is painted all over, then pulled apart. We need how many small cubes show exactly one painted face.

Givens

The stack is 7 cubes across, 7 deep and 7 high.
Every outside surface is painted, including the bottom.

Unknowns

How many small cubes have exactly one painted face.

Constraints

A small cube's painted sides are exactly the sides that faced outward while it was in the stack.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #10 Create a Physical Representation#1 Draw a Diagram

Do not count 343 cubes one at a time. Sort them by where they sit, because position is the only thing that decides how many sides were exposed -- then count positions instead of cubes.

3 · Execute4 carry out the plan

1Sort the small cubes by where they sit

#17 Visualize Spatial Relationships 5.MD.C.4

There are only four kinds of place in the stack: corners (3 sides showing), edges (2), the middle of a face (1), and the hidden core (0). Every small cube is in exactly one kind of place.

3, 2, 1, 0

Position decides the count, not luck.

2One painted face pins down the place

#1 Draw a Diagram 5.MD.C.3

A cube with exactly one painted side had exactly one of its sides facing out, and that happens at a face centre and nowhere else.

\text{face centre} \rightarrow 1

So we only need to look at the face centres.

3Count how many face centres a cube has

#10 Create a Physical Representation 5.MD.C.3

This does not depend on how big the stack is -- any cube has 6 face centres.

6

6 face centres, always.

4Count the cubes in one face centre, then multiply

#17 Visualize Spatial Relationships 5.MD.C.5

With 7 cubes along a side, the middle of a face is that square with its border stripped off -- that is 25 small cubes in one face centre. Multiply that by the 6 face centres.

(7 - 2) \times (7 - 2) = 25

6 x 25 = 150.

Answer: 150

4 · Reviewdoes it hold up?

The four kinds add up to the whole stack: 8 + 12 x 5 + 6 x 25 + 125 = 343, which is 7 x 7 x 7 = 343. Nothing is missing or double-counted.

Another way: Counting cube by cube would also work but takes 343 checks. Sorting by place turns it into one multiplication.

Standardsmin grade 5

5.MD.C.3 Recognize volume as an attribute of solid figures — Reading the stack as a solid whose outside surface is what gets painted.
5.MD.C.4 Measure volumes by counting unit cubes — Counting the small unit cubes by the place they occupy.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Turning the count into places x cubes-per-place.

💡Takeaway. Where a small cube sits is what decides how much paint it gets, so count places instead of cubes.

Where next?

← Back to the problem: Exposed faces depend on a cube's position