← Count everything, then take the overlaps back off · Cube Stacks and Their Views

Count everything, then take the overlaps back off · 12 practice problems

6.G.A.27.G.B.6

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Generated variants — 12

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

Variant 1 easy

A cube is built from $125$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 125 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 5 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 5 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 5 cubes.

5 \times 3 = 15

15 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

15 - 2 = 13

13 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

125 - 13 = 112

112 cubes remain.

Answer: 112 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 110, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 9 cubes and each of the other 4 layers loses 1, which adds to the same 13.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 2 easy

A cube is built from $343$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 343 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 7 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 7 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 7 cubes.

7 \times 3 = 21

21 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

21 - 2 = 19

19 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

343 - 19 = 324

324 cubes remain.

Answer: 324 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 322, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 13 cubes and each of the other 6 layers loses 1, which adds to the same 19.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 3 easy

A cube is built from $729$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 729 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 9 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 9 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 9 cubes.

9 \times 3 = 27

27 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

27 - 2 = 25

25 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

729 - 25 = 704

704 cubes remain.

Answer: 704 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 702, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 17 cubes and each of the other 8 layers loses 1, which adds to the same 25.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 4 easy

A cube is built from $1331$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 1331 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 11 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 11 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 11 cubes.

11 \times 3 = 33

33 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

33 - 2 = 31

31 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

1331 - 31 = 1300

1300 cubes remain.

Answer: 1300 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 1298, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 21 cubes and each of the other 10 layers loses 1, which adds to the same 31.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 5 medium

A cube is built from $2197$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 2197 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 13 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 13 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 13 cubes.

13 \times 3 = 39

39 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

39 - 2 = 37

37 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

2197 - 37 = 2160

2160 cubes remain.

Answer: 2160 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 2158, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 25 cubes and each of the other 12 layers loses 1, which adds to the same 37.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 6 medium

A cube is built from $3375$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 3375 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 15 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 15 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 15 cubes.

15 \times 3 = 45

45 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

45 - 2 = 43

43 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

3375 - 43 = 3332

3332 cubes remain.

Answer: 3332 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 3330, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 29 cubes and each of the other 14 layers loses 1, which adds to the same 43.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 7 medium

A cube is built from $4913$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 4913 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 17 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 17 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 17 cubes.

17 \times 3 = 51

51 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

51 - 2 = 49

49 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

4913 - 49 = 4864

4864 cubes remain.

Answer: 4864 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 4862, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 33 cubes and each of the other 16 layers loses 1, which adds to the same 49.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 8 medium

A cube is built from $6859$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 6859 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 19 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 19 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 19 cubes.

19 \times 3 = 57

57 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

57 - 2 = 55

55 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

6859 - 55 = 6804

6804 cubes remain.

Answer: 6804 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 6802, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 37 cubes and each of the other 18 layers loses 1, which adds to the same 55.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 9 hard

A cube is built from $9261$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 9261 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 21 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 21 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 21 cubes.

21 \times 3 = 63

63 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

63 - 2 = 61

61 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

9261 - 61 = 9200

9200 cubes remain.

Answer: 9200 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 9198, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 41 cubes and each of the other 20 layers loses 1, which adds to the same 61.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 10 hard

A cube is built from $15625$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 15625 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 25 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 25 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 25 cubes.

25 \times 3 = 75

75 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

75 - 2 = 73

73 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

15625 - 73 = 15552

15552 cubes remain.

Answer: 15552 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 15550, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 49 cubes and each of the other 24 layers loses 1, which adds to the same 73.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 11 hard

A cube is built from $29791$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 29791 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 31 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 31 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 31 cubes.

31 \times 3 = 93

93 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

93 - 2 = 91

91 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

29791 - 91 = 29700

29700 cubes remain.

Answer: 29700 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 29698, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 61 cubes and each of the other 30 layers loses 1, which adds to the same 91.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Variant 12 hard

A cube is built from $68921$ unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?

Your answer

Show solution

1 · Understandwhat's really being asked

A cube of 68921 unit cubes has a one-cube line drilled through the middle of each pair of opposite faces. We want how many cubes are left.

Givens

The cube is 41 unit cubes along each edge.
A line of cubes is removed through the middle of each face.

Unknowns

The number of unit cubes remaining.

Constraints

Each line runs right through, so it removes a whole row of 41 cubes.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#12 Draw a Venn Diagram

Counting what is left directly means picturing a shape with three crossing holes. Count what is removed instead -- and then check whether any cube was removed twice.

3 · Execute4 carry out the plan

1Count the lines

#17 Visualize Spatial Relationships 6.G.A.2

Three directions, each a line of 41 cubes.

41 \times 3 = 123

123 cubes, if none is shared.

2Find the overlap

#12 Draw a Venn Diagram 7.G.B.6

All three lines pass through the very middle of the cube, so that one cube sits in every line.

1 \text{ cube, in all } 3

It was counted three times.

3Take the extra countings off

#16 Count the Complement 7.G.B.6

Counting one cube three times means two of those countings were extra.

123 - 2 = 121

121 cubes actually leave.

4Subtract from the whole cube

#16 Count the Complement 6.G.A.2

What is left is everything else.

68921 - 121 = 68800

68800 cubes remain.

Answer: 68800 cubes

4 · Reviewdoes it hold up?

Forgetting the overlap would give 68798, two cubes short -- exactly the two extra countings of the centre cube.

Another way: Counting layer by layer works too: the middle layer loses 81 cubes and each of the other 40 layers loses 1, which adds to the same 121.

Standardsmin grade 7

6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the whole solid.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Accounting for the cube the three removals share.

💡Takeaway. When the things you count overlap, count them all and then take the double counting back off.

Where next?

← Back to the problem: Count everything, then take the overlaps back off