← Opposite faces pair up down a glued stack · Net and Solid Structure

Opposite faces pair up down a glued stack · 12 practice problems

1.OA.D.86.G.A.4

0 / 12 solved

Show all answers (parent / teacher)

Generated variants — 12

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

Variant 1 easy

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Two such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $6$ , find the number of pips on the bottom face that rests on the ground.

The two dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $5$ pips. There are one places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Two dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 6. The top of the tower shows 5. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 6.
The topmost face shows 5 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 6 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute3 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 5. Its bottom face is the opposite face, so it must make 7 with it.

7 - 5 = 2

Underneath this die is 2.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (2) touches the top of this one, and a touching pair sums to 6. So this die's top face is 6 - 2.

6 - 2 = 4

This die shows 4 on top.

3Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 4, so the face resting on the ground is the opposite face, making 7 with it.

7 - 4 = 3

The ground face shows 3.

Answer: 3

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 5, 2, 4, 3 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is 1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 6. Starting at 2, 1 steps of 1 land on 3.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-6 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 2 easy

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Two such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $10$ , find the number of pips on the bottom face that rests on the ground.

The two dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $1$ pips. There are one places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Two dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 10. The top of the tower shows 1. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 10.
The topmost face shows 1 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 10 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute3 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 1. Its bottom face is the opposite face, so it must make 7 with it.

7 - 1 = 6

Underneath this die is 6.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (6) touches the top of this one, and a touching pair sums to 10. So this die's top face is 10 - 6.

10 - 6 = 4

This die shows 4 on top.

3Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 4, so the face resting on the ground is the opposite face, making 7 with it.

7 - 4 = 3

The ground face shows 3.

Answer: 3

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 1, 6, 4, 3 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -3 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 10. Starting at 6, 1 steps of -3 land on 3.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-10 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 3 easy

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Two such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $11$ , find the number of pips on the bottom face that rests on the ground.

The two dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $2$ pips. There are one places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Two dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 11. The top of the tower shows 2. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 11.
The topmost face shows 2 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 11 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute3 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 2. Its bottom face is the opposite face, so it must make 7 with it.

7 - 2 = 5

Underneath this die is 5.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (5) touches the top of this one, and a touching pair sums to 11. So this die's top face is 11 - 5.

11 - 5 = 6

This die shows 6 on top.

3Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 6, so the face resting on the ground is the opposite face, making 7 with it.

7 - 6 = 1

The ground face shows 1.

Answer: 1

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 2, 5, 6, 1 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -4 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 11. Starting at 5, 1 steps of -4 land on 1.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-11 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 4 easy

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Three such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $6$ , find the number of pips on the bottom face that rests on the ground.

The three dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $4$ pips. There are two places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Three dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 6. The top of the tower shows 4. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 6.
The topmost face shows 4 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 6 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute5 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 4. Its bottom face is the opposite face, so it must make 7 with it.

7 - 4 = 3

Underneath this die is 3.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (3) touches the top of this one, and a touching pair sums to 6. So this die's top face is 6 - 3.

6 - 3 = 3

This die shows 3 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 3. Its bottom face is the opposite face, so it must make 7 with it.

7 - 3 = 4

Underneath this die is 4.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (4) touches the top of this one, and a touching pair sums to 6. So this die's top face is 6 - 4.

6 - 4 = 2

This die shows 2 on top.

5Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 2, so the face resting on the ground is the opposite face, making 7 with it.

7 - 2 = 5

The ground face shows 5.

Answer: 5

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 4, 3, 3, 4, 2, 5 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is 1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 6. Starting at 3, 2 steps of 1 land on 5.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-6 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 5 medium

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Three such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $8$ , find the number of pips on the bottom face that rests on the ground.

The three dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $4$ pips. There are two places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Three dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 8. The top of the tower shows 4. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 8.
The topmost face shows 4 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 8 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute5 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 4. Its bottom face is the opposite face, so it must make 7 with it.

7 - 4 = 3

Underneath this die is 3.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (3) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 3.

8 - 3 = 5

This die shows 5 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 5. Its bottom face is the opposite face, so it must make 7 with it.

7 - 5 = 2

Underneath this die is 2.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (2) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 2.

8 - 2 = 6

This die shows 6 on top.

5Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 6, so the face resting on the ground is the opposite face, making 7 with it.

7 - 6 = 1

The ground face shows 1.

Answer: 1

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 4, 3, 5, 2, 6, 1 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 8. Starting at 3, 2 steps of -1 land on 1.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-8 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 6 medium

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Three such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $8$ , find the number of pips on the bottom face that rests on the ground.

The three dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $2$ pips. There are two places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Three dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 8. The top of the tower shows 2. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 8.
The topmost face shows 2 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 8 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute5 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 2. Its bottom face is the opposite face, so it must make 7 with it.

7 - 2 = 5

Underneath this die is 5.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (5) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 5.

8 - 5 = 3

This die shows 3 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 3. Its bottom face is the opposite face, so it must make 7 with it.

7 - 3 = 4

Underneath this die is 4.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (4) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 4.

8 - 4 = 4

This die shows 4 on top.

5Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 4, so the face resting on the ground is the opposite face, making 7 with it.

7 - 4 = 3

The ground face shows 3.

Answer: 3

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 2, 5, 3, 4, 4, 3 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 8. Starting at 5, 2 steps of -1 land on 3.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-8 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 7 medium

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Three such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $8$ , find the number of pips on the bottom face that rests on the ground.

The three dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $3$ pips. There are two places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Three dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 8. The top of the tower shows 3. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 8.
The topmost face shows 3 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 8 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute5 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 3. Its bottom face is the opposite face, so it must make 7 with it.

7 - 3 = 4

Underneath this die is 4.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (4) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 4.

8 - 4 = 4

This die shows 4 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 4. Its bottom face is the opposite face, so it must make 7 with it.

7 - 4 = 3

Underneath this die is 3.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (3) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 3.

8 - 3 = 5

This die shows 5 on top.

5Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 5, so the face resting on the ground is the opposite face, making 7 with it.

7 - 5 = 2

The ground face shows 2.

Answer: 2

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 3, 4, 4, 3, 5, 2 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 8. Starting at 4, 2 steps of -1 land on 2.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-8 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 8 hard

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Three such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $9$ , find the number of pips on the bottom face that rests on the ground.

The three dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $2$ pips. There are two places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Three dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 9. The top of the tower shows 2. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 9.
The topmost face shows 2 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 9 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute5 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 2. Its bottom face is the opposite face, so it must make 7 with it.

7 - 2 = 5

Underneath this die is 5.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (5) touches the top of this one, and a touching pair sums to 9. So this die's top face is 9 - 5.

9 - 5 = 4

This die shows 4 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 4. Its bottom face is the opposite face, so it must make 7 with it.

7 - 4 = 3

Underneath this die is 3.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (3) touches the top of this one, and a touching pair sums to 9. So this die's top face is 9 - 3.

9 - 3 = 6

This die shows 6 on top.

5Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 6, so the face resting on the ground is the opposite face, making 7 with it.

7 - 6 = 1

The ground face shows 1.

Answer: 1

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 2, 5, 4, 3, 6, 1 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -2 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 9. Starting at 5, 2 steps of -2 land on 1.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-9 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 9 medium

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Three such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $9$ , find the number of pips on the bottom face that rests on the ground.

The three dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $1$ pips. There are two places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Three dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 9. The top of the tower shows 1. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 9.
The topmost face shows 1 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 9 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute5 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 1. Its bottom face is the opposite face, so it must make 7 with it.

7 - 1 = 6

Underneath this die is 6.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (6) touches the top of this one, and a touching pair sums to 9. So this die's top face is 9 - 6.

9 - 6 = 3

This die shows 3 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 3. Its bottom face is the opposite face, so it must make 7 with it.

7 - 3 = 4

Underneath this die is 4.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (4) touches the top of this one, and a touching pair sums to 9. So this die's top face is 9 - 4.

9 - 4 = 5

This die shows 5 on top.

5Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 5, so the face resting on the ground is the opposite face, making 7 with it.

7 - 5 = 2

The ground face shows 2.

Answer: 2

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 1, 6, 3, 4, 5, 2 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -2 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 9. Starting at 6, 2 steps of -2 land on 2.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-9 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 10 hard

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Four such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $8$ , find the number of pips on the bottom face that rests on the ground.

The four dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $3$ pips. There are three places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Four dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 8. The top of the tower shows 3. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 8.
The topmost face shows 3 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 8 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute7 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 3. Its bottom face is the opposite face, so it must make 7 with it.

7 - 3 = 4

Underneath this die is 4.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (4) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 4.

8 - 4 = 4

This die shows 4 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 4. Its bottom face is the opposite face, so it must make 7 with it.

7 - 4 = 3

Underneath this die is 3.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (3) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 3.

8 - 3 = 5

This die shows 5 on top.

5Find the bottom of the third die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 5. Its bottom face is the opposite face, so it must make 7 with it.

7 - 5 = 2

Underneath this die is 2.

6Cross to the fourth die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (2) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 2.

8 - 2 = 6

This die shows 6 on top.

7Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 6, so the face resting on the ground is the opposite face, making 7 with it.

7 - 6 = 1

The ground face shows 1.

Answer: 1

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 3, 4, 4, 3, 5, 2, 6, 1 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 8. Starting at 4, 3 steps of -1 land on 1.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-8 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 11 hard

The four dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $1$ pips. There are three places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Four dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 8. The top of the tower shows 1. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 8.
The topmost face shows 1 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 8 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute7 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 1. Its bottom face is the opposite face, so it must make 7 with it.

7 - 1 = 6

Underneath this die is 6.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (6) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 6.

8 - 6 = 2

This die shows 2 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 2. Its bottom face is the opposite face, so it must make 7 with it.

7 - 2 = 5

Underneath this die is 5.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (5) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 5.

8 - 5 = 3

This die shows 3 on top.

5Find the bottom of the third die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 3. Its bottom face is the opposite face, so it must make 7 with it.

7 - 3 = 4

Underneath this die is 4.

6Cross to the fourth die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (4) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 4.

8 - 4 = 4

This die shows 4 on top.

7Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 4, so the face resting on the ground is the opposite face, making 7 with it.

7 - 4 = 3

The ground face shows 3.

Answer: 3

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 1, 6, 2, 5, 3, 4, 4, 3 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 8. Starting at 6, 3 steps of -1 land on 3.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-8 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Variant 12 hard

On each die, the two faces that are opposite each other have pip counts that add up to $7$ . Five such dice are glued together as shown at the right to make a rectangular prism. If the pip counts on each pair of touching faces add up to $8$ , find the number of pips on the bottom face that rests on the ground.

The five dice are stacked straight up to form the rectangular prism. The top face of the topmost die shows $2$ pips. There are four places where the faces of stacked dice touch each other.

Your answer

Show solution

1 · Understandwhat's really being asked

Five dice are glued into a tower. Opposite faces of a die add to 7, and every glued pair adds to 8. The top of the tower shows 2. We need the number on the very bottom.

Givens

On one die, opposite faces add to 7.
Each pair of glued faces adds to 8.
The topmost face shows 2 pips.

Unknowns

The number of pips on the face resting on the ground.

Constraints

Every face value is between 1 and 6.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#6 Guess and Check

Two rules take turns going down the tower: 7 to get from a die's top to its own bottom, and 8 to get across a glued join. Apply them alternately from the top and the last one lands on the ground.

3 · Execute9 carry out the plan

1Find the bottom of the top die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 2. Its bottom face is the opposite face, so it must make 7 with it.

7 - 2 = 5

Underneath this die is 5.

2Cross to the second die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (5) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 5.

8 - 5 = 3

This die shows 3 on top.

3Find the bottom of the second die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 3. Its bottom face is the opposite face, so it must make 7 with it.

7 - 3 = 4

Underneath this die is 4.

4Cross to the third die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (4) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 4.

8 - 4 = 4

This die shows 4 on top.

5Find the bottom of the third die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 4. Its bottom face is the opposite face, so it must make 7 with it.

7 - 4 = 3

Underneath this die is 3.

6Cross to the fourth die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (3) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 3.

8 - 3 = 5

This die shows 5 on top.

7Find the bottom of the fourth die

#10 Create a Physical Representation 6.G.A.4

This die's top face shows 5. Its bottom face is the opposite face, so it must make 7 with it.

7 - 5 = 2

Underneath this die is 2.

8Cross to the fifth die

#6 Guess and Check 1.OA.D.8

The bottom of the die above (2) touches the top of this one, and a touching pair sums to 8. So this die's top face is 8 - 2.

8 - 2 = 6

This die shows 6 on top.

9Find the ground face

#1 Draw a Diagram 6.G.A.4

This die's top is 6, so the face resting on the ground is the opposite face, making 7 with it.

7 - 6 = 1

The ground face shows 1.

Answer: 1

4 · Reviewdoes it hold up?

Every face on the way is a real die face -- 2, 5, 3, 4, 4, 3, 5, 2, 6, 1 -- and each is between 1 and 6, so nothing impossible happened on the way down.

Another way: Notice the pattern: each die's bottom is -1 more than the one above it, because crossing a die adds 7 and crossing a join subtracts 8. Starting at 5, 4 steps of -1 land on 1.

Standardsmin grade 6

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation — Filling the missing number in each add-to-7 and add-to-8 pair.
6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Tracking which faces are opposite each other on a solid.

💡Takeaway. Going down the tower you only ever cross two kinds of pair -- opposite faces and glued faces -- and each has its own total.

Where next?

← Back to the problem: Opposite faces pair up down a glued stack