← 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

Generated variants — 12

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

Variant 1 easy answer: 112 cubes

A cube is built from 125125 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?

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.
5×3=155 \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 cube, in all 31 \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=1315 - 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=112125 - 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 answer: 324 cubes

A cube is built from 343343 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?

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.
7×3=217 \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 cube, in all 31 \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=1921 - 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=324343 - 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 answer: 704 cubes

A cube is built from 729729 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?

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.
9×3=279 \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 cube, in all 31 \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=2527 - 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=704729 - 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 answer: 1300 cubes

A cube is built from 13311331 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?

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.
11×3=3311 \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 cube, in all 31 \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=3133 - 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=13001331 - 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 answer: 2160 cubes

A cube is built from 21972197 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?

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.
13×3=3913 \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 cube, in all 31 \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=3739 - 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=21602197 - 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 answer: 3332 cubes

A cube is built from 33753375 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?

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.
15×3=4515 \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 cube, in all 31 \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=4345 - 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=33323375 - 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 answer: 4864 cubes

A cube is built from 49134913 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?

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.
17×3=5117 \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 cube, in all 31 \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=4951 - 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=48644913 - 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 answer: 6804 cubes

A cube is built from 68596859 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?

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.
19×3=5719 \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 cube, in all 31 \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=5557 - 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=68046859 - 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 answer: 9200 cubes

A cube is built from 92619261 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?

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.
21×3=6321 \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 cube, in all 31 \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=6163 - 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=92009261 - 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 answer: 15552 cubes

A cube is built from 1562515625 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?

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.
25×3=7525 \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 cube, in all 31 \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=7375 - 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=1555215625 - 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 answer: 29700 cubes

A cube is built from 2979129791 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?

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.
31×3=9331 \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 cube, in all 31 \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=9193 - 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=2970029791 - 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 answer: 68800 cubes

A cube is built from 6892168921 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?

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.
41×3=12341 \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 cube, in all 31 \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=121123 - 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=6880068921 - 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.