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Cube & Dice

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medium importance~0-1 Q in Tier 16 formulas⚡ 7 shortcuts3 subtopics
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Dice: finding opposite faces from positions

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⏱ 5 min read🧩 7 question types🎯 18 practice Q
The idea in one minute

A die is a cube with six faces. Each face touches four faces and is opposite exactly one face. When several pictures of the same die are shown, use them to find which faces are opposite. Three quick rules do most of the work: strike out the neighbours, look for two common faces, and compare faces that sit in the same place.

01

What one picture shows

A die has six faces. Each face touches four faces and is opposite one face.

A picture shows three faces that meet at one corner. These three faces all touch each other.

Rule: Two faces seen together in one picture are never opposite.

02

Strike out the neighbours

To find the face opposite X, list every face seen together with X. All of them touch X.

If four different faces are struck out, the one face left is opposite X.

Example: X is seen with A and B in one picture, with C in a second picture, and with D in a third. A, B, C and D all touch X. So the sixth face is opposite X.

Tip: This rule needs no turning of the die. Try it first.

03

Two common faces

Sometimes two pictures show the same two faces. Then the third face in each picture are opposite each other.

Pictures (K,L,M)(K, L, M) and (K,L,N)(K, L, N) share K and L. So M and N are opposite.

Why? The two faces that touch both K and L are exactly the two faces at the ends of the shared edge. They are opposite each other.

04

One common face in the same place

Now two pictures share only one face X. X sits in the same place in both, for example on top. All other faces are different.

Then the faces in the same place are opposite: front with front, right with right.

Example: picture 1 has top A, front B, right C. Picture 2 has top A, front D, right E. So B is opposite D, C is opposite E, and F is opposite A.

Watch: Use this rule only when the common face is in the same place and no other face repeats. If not, use the first two rules.

05

Standard die

A standard (ordinary) die has opposite faces adding to 7: 1 and 6, 2 and 5, 3 and 4.

All six faces add to 21. So the hidden faces add to 2121 minus the three faces you can see.

Note: Use the sum 7 rule only when the question says the die is standard. A die marked with letters has no sum rule.

06

Rolling a die

Rolling a die moves its faces in a circle. The other two faces stay in place.

  • Roll to the right: top goes to the right, right goes to the bottom, bottom goes to the left, left comes to the top.
  • Roll forward: top goes to the front, front goes to the bottom, bottom goes to the back, back comes to the top.

Example: a standard die has 1 on top, 2 in front and 3 on the right. So 6 is at the bottom, 5 at the back and 4 on the left. Roll it once to the right: 4 comes to the top.

07

Order of work

Write the three opposite pairs as you find them.

  1. Look for two common faces. This gives a whole pair at once.
  2. Look for a common face in the same place.
  3. Strike out neighbours to finish the last pair.
08

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Strike out the neighbours

How to spot it:

Three pictures of one die. The asked face appears with four different faces across the pictures.

Method

  1. List every face seen together with the asked face.

  2. Strike them out. They all touch it.

  3. The one face left is the answer.

Why it works:

Every face seen with X touches X, so the only face never seen with X must be opposite.

Try this

Three pictures of one cube show the faces (P, Q, R), (P, S, Q) and (P, R, T). Which face is opposite P?

Show solution

  1. P is seen with Q, R, S and T.

  2. These four faces all touch P.

  3. Only U is left.

Answer

U

Type 2very common2 practice Q

Two common faces

How to spot it:

Two pictures repeat the same two faces. Each picture has one extra face.

Method

  1. Find the two faces that appear in both pictures.

  2. Note the extra face in each picture.

  3. The two extra faces are opposite.

Why it works:

Only two faces touch both shared faces. They sit at the two ends of the shared edge.

Try this

One picture of a cube shows the faces K, L and M. Another picture shows K, L and N. Which face is opposite M?

Show solution

  1. K and L are in both pictures.

  2. The extra faces are M and N.

  3. So M and N are opposite.

Answer

N

Type 3common2 practice Q

One common face in the same place

How to spot it:

Two pictures share exactly one face. It sits in the same place in both, such as top in both.

Method

  1. Check that the common face is in the same place.

  2. Check that no other face repeats.

  3. Pair the faces in matching places: front with front, right with right.

Why it works:

The four faces round the common face go in a circle, so faces in matching places end up across from each other.

Try this

In picture 1 a cube shows A on top, B in front and C on the right. In picture 2 it shows A on top, D in front and E on the right. Which face is opposite B?

Show solution

  1. A is on top in both pictures.

  2. No other face repeats.

  3. Front faces are opposite: B and D.

  4. (Right faces C and E are opposite. F is opposite A.)

Answer

D

Type 4common2 practice Q

Standard die (sum 7)

How to spot it:

The question says ordinary or standard die and gives some face numbers.

x+x′=7x + x' = 7
Method

  1. Pair the numbers: 1-6, 2-5, 3-4.

  2. For a hidden face, take 7 minus its opposite.

  3. For the hidden total, take 21 minus the shown faces.

Why it works:

Opposite faces add to 7, and all six faces add to 21.

Try this

A standard die shows 3 on top, 5 in front and 6 on the right. What is the sum of the three hidden faces?

Show solution

  1. Shown: 3+5+6=143+5+6=14.

  2. Hidden: 21−14=721-14=7.

  3. Check: the hidden faces are 4, 2 and 1.

Answer

7

Type 5occasional2 practice Q

Build the full opposite map

How to spot it:

Three or more pictures. The question asks for one pair or for the partner of the last face.

Method

  1. Use two common faces to find a pair.

  2. Repeat with another two pictures.

  3. The last two unpaired faces are opposite.

Why it works:

There are only three opposite pairs. Two known pairs leave the third.

Try this

Three pictures of a cube show (A, B, C), (A, B, D) and (B, C, E). Which face is opposite B?

Show solution

  1. First and second pictures share A and B, so C is opposite D.

  2. First and third pictures share B and C, so A is opposite E.

  3. Only B and F are left, so B is opposite F.

Answer

F

Type 6occasional2 practice Q

Which picture is possible?

How to spot it:

The opposite pairs are given in words. The options are pictures of three faces.

Method

  1. Write the three opposite pairs.

  2. Cross out any option that shows both faces of a pair.

  3. The option with one face from each pair is possible.

Why it works:

A picture never shows two opposite faces. Three faces from three different pairs always meet at a corner.

Try this

On a cube, A is opposite D, B is opposite E and C is opposite F. Which set of three faces can be seen together? (1) A, D, C (2) B, E, F (3) A, B, F (4) C, F, B

Show solution

  1. Option 1 has A and D. Option 2 has B and E.

  2. Option 4 has C and F.

  3. Option 3 has A, B and F, one from each pair.

Answer

Option 3

Type 7common

Rolling the die

How to spot it:

The die is rolled to the right, left, forward or back. The top, bottom or front face is asked.

Method

  1. Write the six faces: top, bottom, front, back, left, right.

  2. Roll right: left face comes to the top.

  3. Roll forward: back face comes to the top.

  4. Write the new positions after every roll.

Why it works:

A roll turns the die by a quarter turn, so four faces move in a circle and two stay.

Try this

A standard die shows 1 on top, 2 in front and 3 on the right. It is rolled once to the right and then once forward. Which number is now on top?

Show solution

  1. Bottom 6, back 5, left 4 (each pair adds to 7).

  2. Roll right: left face 4 comes to the top.

  3. Now the back face is still 5.

  4. Roll forward: the back face 5 comes to the top.

Answer

5

09

Formula sheet

Standard die
1+6=2+5=3+4=71+6 = 2+5 = 3+4 = 7

Opposite faces of a standard die add to 7.

Hidden faces of a standard die
hidden=21−shown\text{hidden} = 21 - \text{shown}

All six faces add to 21.

10

Shortcuts that save time

⚡ Strike out the neighbours

List every face seen with the asked face and cross them out. If one face is left, it is the answer.

Example

A face P is seen with Q and R, then with S and Q, then with R and T. Which face is opposite P?

Show solution

  1. P was seen with Q, R, S and T.

  2. These four all touch P.

  3. Only U is left.

Answer

U

⚡ Two common faces

Two pictures with two faces in common: the two other faces are opposite.

Example

One picture shows K, L, M. Another shows K, L, N. Which face is opposite M?

Show solution

  1. K and L appear in both pictures.

  2. The other faces, M and N, are opposite each other.

Answer

N

⚡ Hidden faces of a standard die

All six faces add to 21. Subtract the three faces you can see.

Example

A standard die shows 2, 4 and 6. What do the three hidden faces add up to?

Show solution

  1. Shown: 2+4+6=122+4+6=12.

  2. Hidden: 21−12=921-12=9.

  3. Check: the hidden faces are 5, 3 and 1, and 5+3+1=95+3+1=9.

Answer

9

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Saying two faces are opposite only because they never appear together.

First strike out four neighbours, or use two common faces. Then the last face is opposite.

Mistake 02

Using the same-place rule when the common face is in a different place, or when a second face also repeats.

Use it only for one common face in the same place. Otherwise use strike-out or two common faces.

Mistake 03

Using the sum 7 rule on a die that is not called standard.

Check the question. Letters or unusual numbers have no sum rule.

Mistake 04

Choosing an option that shows two opposite faces together.

Reject any picture that shows two faces from the same opposite pair.

Mistake 05

Mixing up left and right when rolling a die.

Rolling right sends the top to the right side and brings the left face up. Say it aloud.

12

Quick revision

Read this the night before the exam.

  • A picture shows three faces at one corner. They all touch.

  • Strike out every face seen with X. Four struck out means the sixth is opposite.

  • Two common faces: the other two faces are opposite.

  • One common face in the same place: front with front, right with right.

  • Standard die: 1-6, 2-5, 3-4. Hidden faces add to 21 minus the shown faces.

  • Roll right: left face comes to the top. Roll forward: back face comes to the top.

13

Practice: 18 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.

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