Cube & Dice
🔒 Log in to trackDice: finding opposite faces from positions
🔒 Log in to trackA die (cube) has 6 faces: each face touches 4 others and is opposite exactly 1. When several views of the same die are shown, use these rules:
- Adjacent-four rule — a face seen next to four different faces is opposite the sixth one.
- Two common faces — if two views share two faces, the remaining face in each view is opposite the other one.
- Same position — if one face is common and it sits in the same position in both views, the faces in matching positions are opposite (front ↔ front, right ↔ right).
- Never together — two faces seen together in any view can never be opposite.
A standard die has opposite faces summing to 7: (1, 6), (2, 5), (3, 4).
Detailed notes
What a view of a die tells you
A die has six faces, and each face has exactly one face opposite it; the other four touch it. A view of a die shows three faces that meet at one corner, so the three faces in any one view are pairwise adjacent — no two of them can ever be opposite.
Rule 1: strike out the faces seen together
To find the face opposite X, list every face that appears in some view together with X. All of them touch X. If four different faces are struck out, exactly one face is left, and that face must be opposite X. This single rule settles most questions. Example: X appears with A and B in one view, with C in a second and with D in a third. Faces A, B, C, D all touch X, so the sixth face is opposite X.
Rule 2: two common faces
If two views share the same two faces, then the face left over in the first view is opposite the face left over in the second. Views (A, B, C) and (A, B, D): A and B are common, so C and D are opposite.
Rule 3: common face in the same position
If one face is common to two views and sits in the same position in both (top in both, front in both, or right in both), then the faces in the other matching positions are opposite: front of view 1 with front of view 2, right with right. If the common face has moved to a different position, this rule does not apply — fall back on Rules 1 and 2.
Rule 4: never together
Two faces that appear together in any single view can never be opposite. This is the quickest way to reject an option or a suggested pair.
Standard die
A standard (ordinary) die has opposite faces adding to 7: 1 is opposite 6, 2 opposite 5, 3 opposite 4. Use this only when the question says the die is standard; a die showing letters has no sum rule.
The working method
Keep a small table of opposite pairs and fill it in as you read the views. A die has exactly three opposite pairs, and each face is used once. A good order: apply Rule 2 first (it gives a whole pair at once), then Rule 3, then finish with Rule 1.
Worked example
Three views show (P, Q, R), (R, S, P) and (Q, T, R), where the first letter is the front, second the top, third the right face. Views 1 and 2 share P and R, so the leftovers Q and S are opposite. Views 1 and 3 share Q and R in the same positions (front and right), so the matching positions give P opposite T. That leaves R opposite the sixth face U. All three pairs found.
Common mistakes
- Calling two faces opposite just because they never appear together — you must strike out four faces first, or use Rule 2 or 3.
- Using Rule 3 when the common face sits in different positions in the two views.
- Applying the sum-7 rule to a die that is not stated to be standard.
- Forgetting that the answer must be a face of the same die: it must never appear together with the asked face.
Quick revision
- A view shows three faces meeting at a corner; they are all adjacent.
- Rule 1: strike out every face seen with X; if four are gone, the last one is opposite.
- Rule 2: two common faces in two views → the two leftovers are opposite.
- Rule 3: common face in the same position → matching positions are opposite.
- Faces seen together are never opposite.
- Standard die: 1–6, 2–5, 3–4 (each pair sums to 7).
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Adjacent-four (strike out)very common2 practice Q
Three views of the same die; the asked face appears with four different faces across the views.
Every face seen with X touches X. Strike out all of them; the only face never seen with X is opposite it.
Example: Q. 4 is seen with 1 and 2, then with 3 and 1, then with 5 and 3. What is opposite 4?
4 has been seen with 1, 2, 3 and 5 — four faces. The only one left is 6.
Type 2: Two common facesvery common2 practice Q
Two views repeat the same two faces; each view has one extra face.
The two common faces touch both extras, so the two leftover faces are opposite each other.
Example: Q. Views show (A, B, C) and (A, B, D). What is opposite C?
A and B are common to both views, so D is opposite C.
Type 3: Common face in the same positioncommon2 practice Q
Two views share exactly one face, and it sits in the same slot (both top, both front or both right).
Matching positions hold opposite faces: front with front, top with top, right with right.
Example: Q. In both views A is on top; the front faces are B and C. What is opposite B?
A is common and in the same position, so the front faces are opposite: C.
Type 4: Standard die (sum 7)common2 practice Q
The question says 'ordinary' or 'standard' die and gives one or more face values.
Opposite faces add to 7: 1–6, 2–5, 3–4. The three hidden faces sum to 21 minus the three shown.
Example: Q. A standard die shows 3 on top, 5 in front, 6 on the right. Sum of the hidden faces?
Hidden faces are 4, 2 and 1: 21 − (3 + 5 + 6) = 7.
Type 5: Build the full opposite mapoccasional2 practice Q
Three or more views; the question asks for a pair of opposite faces, or for the partner of the last face.
Collect pairs with Rules 2 and 3 until two pairs are known; the two faces never yet paired form the third pair.
Example: Q. From the views you get A opposite D and B opposite E. What is opposite C?
Only C and F are unpaired, so F.
Type 6: Which view is possible?occasional2 practice Q
The opposite pairs of a labelled die are given in words; options are drawings of views.
A real view never shows two opposite faces. Reject every option that shows a pair; a view with one face from each pair can always be turned up.
Example: Q. A is opposite D, B opposite E, C opposite F. Which view can be formed?
The view showing A, B and C — one face from each opposite pair.
Formulas
Opposite faces of a standard die.
Shortcut tricks
⚡ Eliminate the neighbours
List every face seen with the asked face; strike them out. If only one face is left, it is the answer — no rotation needed.
Example: Q. 4 is seen with 1, 2, 3 and 5. What is opposite 4?
Only 6 is never seen with 4, so 6.
⚡ Two common faces
Two views sharing two faces: the leftover faces are opposite.
Example: Q. Views show (A, B, C) and (A, B, D). Opposite of C?
A and B are common, so D.
Where students lose marks
Declaring two faces opposite just because they never appear together — check that the other four are all ruled out.
Applying the same-position rule when the common face is in different positions.
Forgetting that a standard-die fact (sum 7) only holds when the question says the die is standard.
Practice sets — 18 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 6 min · wrong answers go to your mistake notebook automatically.