← 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

Generated variants — 12

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

Variant 1 easy answer: 3

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 66, 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 55 pips. There are one places where the faces of stacked dice touch each other.

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=27 - 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=46 - 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=37 - 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 answer: 3

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 1010, 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 11 pips. There are one places where the faces of stacked dice touch each other.

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=67 - 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=410 - 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=37 - 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 answer: 1

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 1111, 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 22 pips. There are one places where the faces of stacked dice touch each other.

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=57 - 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=611 - 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=17 - 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 answer: 5

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 66, 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 44 pips. There are two places where the faces of stacked dice touch each other.

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=37 - 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=36 - 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=47 - 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=26 - 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=57 - 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 answer: 1

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 88, 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 44 pips. There are two places where the faces of stacked dice touch each other.

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=37 - 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=58 - 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=27 - 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=68 - 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=17 - 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 answer: 3

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 88, 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 22 pips. There are two places where the faces of stacked dice touch each other.

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=57 - 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=38 - 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=47 - 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=48 - 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=37 - 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 answer: 2

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 88, 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 33 pips. There are two places where the faces of stacked dice touch each other.

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=47 - 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=48 - 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=37 - 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=58 - 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=27 - 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 answer: 1

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 99, 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 22 pips. There are two places where the faces of stacked dice touch each other.

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=57 - 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=49 - 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=37 - 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=69 - 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=17 - 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 answer: 2

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 99, 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 11 pips. There are two places where the faces of stacked dice touch each other.

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=67 - 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=39 - 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=47 - 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=59 - 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=27 - 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 answer: 1

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 88, 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 33 pips. There are three places where the faces of stacked dice touch each other.

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=47 - 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=48 - 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=37 - 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=58 - 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=27 - 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=68 - 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=17 - 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 answer: 3

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 88, 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 11 pips. There are three places where the faces of stacked dice touch each other.

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=67 - 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=28 - 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=57 - 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=38 - 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=47 - 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=48 - 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=37 - 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 answer: 1

On each die, the two faces that are opposite each other have pip counts that add up to 77. 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 88, 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 22 pips. There are four places where the faces of stacked dice touch each other.

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=57 - 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=38 - 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=47 - 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=48 - 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=37 - 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=58 - 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=27 - 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=68 - 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=17 - 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.