Six letters I, O, A, Y, Z and X are written on different faces of a dice. Two positions of this dice are shown in the figures below. Find the letter on the face opposite to O.

Correct — B. I sits on top in both views, so the die has only been spun about its vertical axis.

That makes O, A, Y and Z the four faces around the side, and X — which never shows — the face under I.

Rule: in each view the left visible face and the right visible face are neighbours in the side ring, read in the same direction.

View one gives O then A. View two gives Y then Z. Placing O then A first leaves Y then Z for the remaining slots, so the ring is O → A → Y → Z → O.

In a ring of four, faces two apart are opposite, so O faces Y — option (b).

Why the others are wrong

• (a)
  Z — Z lies two places from A in that ring, not from O, so Z is opposite A. O and Z are neighbours, and neighbours are never an opposite pair.
• (c)
  A — O and A are both visible in the first view. Two faces you can see at the same time are adjacent, so A is ruled out before any rotation work begins.
• (d)
  X — X shows in neither view, which is what makes it the face opposite I, the letter that shows in both. It cannot also be opposite O.

A dice item with two views and one repeated letter is the friendliest kind, because the repeated letter fixes the axis of rotation.

Here I is on top in both views, so nothing has moved except a spin about the vertical axis. The four side faces therefore form one ring, and the sixth letter must lie underneath.

Reading each view in the same order — left visible face, then right visible face — keeps the direction of travel round that ring constant.

Faces two apart in a ring of four are opposite, which converts the ring straight into the answer.

Key facts

• Two faces visible in the same view of a die are adjacent, never opposite.
• A face that appears in both views means the die was rotated about the axis through that face.
• The four faces around a fixed top face form a ring, and faces two apart in it are opposite.
• A die has six faces in three opposite pairs, so fixing one pair narrows the rest.

Common traps

• Assuming the two views show the same three faces in a different orientation.
• Reading one view left to right and the other right to left, which reverses the ring.
• Choosing a letter that shares a view with O, when a shared view proves adjacency.

No directly related past PYQ was found.