A cube whose faces are marked with letters is shown below in different…
2013
A cube whose faces are marked with letters is shown below in different positions as viewed from different directions. Find the missing letter indicated by the question mark (?).
- A.
S
- B.
D
- C.
Y
- D.
M
Attempted by 3 students.
Show answer & explanation
Correct answer: C
On a cube, the six faces form exactly three opposite pairs, and any two faces that appear together in a single view — its top, front and right, say — are, by definition, adjacent, never opposite.
So across several rotated views of the same cube, every pair of letters ever seen together on one view is ruled out as an opposite pair; comparing enough views, together with tracking how faces move under a rotation, pins down every letter's one valid opposite partner.
Apply this to the four given views of the cube:
Cube | Top | Front | Right |
|---|---|---|---|
Cube 1 | D | M | S |
Cube 2 | S | M | W |
Cube 3 | Y | U | D |
Cube 4 | W | U | ? |
Work through the deduction:
Cube 1 shows D, M and S together, so no two of D, M and S can be opposite each other.
Cube 2 shows S, M and W together, so no two of S, M and W can be opposite each other.
Cube 3 shows Y, U and D together, so no two of Y, U and D can be opposite each other.
D is therefore ruled out as the opposite of M and S (from Cube 1) and of U and Y (from Cube 3) — the only letter left for D is W, so D is opposite W.
This adjacency check alone still leaves two possibilities for the remaining letters M, S, Y, U — either (M opposite Y, S opposite U) or (M opposite U, S opposite Y) — since only M-S and Y-U are directly ruled out. Cube 3 and Cube 4 share the same front face, U, so Cube 4 is simply Cube 3 turned about that front-back axis; tracking this turn (not adjacency alone) settles which pairing is correct.
In Cube 3, the hidden left face is opposite the visible right face (D), which is W. So, around the U-axis, Cube 3's faces run Top = Y, Right = D, Left = W, in that rotational order.
Turning Cube 3 about this same axis until its Left face (W) comes to the Top — exactly Cube 4's view — carries Cube 3's old Top face (Y) around into the new Right position. So Cube 4's right face is Y, which fixes M as Y's opposite (not U).
As a further check: Cube 1 and Cube 2 also share one face, M (front); turning Cube 1 about that axis carries its right face (S) onto Cube 2's top face (S) exactly as required, and the same turn carries Cube 1's hidden bottom face onto Cube 2's right face (W) — so Cube 1's bottom face is W. Since top and bottom are always opposite, this independently confirms D is opposite W, matching the elimination above.
Hence the missing letter marked by the question mark is Y.