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Difficulty: MediumCubes and Dice

Three different positions of the same die are shown below. The six faces of the die are marked with the Greek letters α,β,γ,δ,ϵ\alpha, \beta, \gamma, \delta, \epsilon, and ζ\zeta.

- Position 1: Top is α\alpha, Front is β\beta, Right is δ\delta
- Position 2: Top is β\beta, Front is γ\gamma, Right is δ\delta
- Position 3: Top is β\beta, Front is α\alpha, Right is ζ\zeta

Based on the given positions, which letter is on the face opposite to δ\delta?

  1. A
    α\alpha
  2. B
    β\beta
  3. C
    γ\gamma
  4. ζ\zetaAnswer

Answer

The letter on the face opposite to δ\delta is ζ\zeta.
By analyzing the adjacent faces across the three positions, we can determine all opposite pairs. Gathering all unique adjacent faces for β\beta, we find it is adjacent to α,γ,δ\alpha, \gamma, \delta, and ζ\zeta. This leaves ϵ\epsilon as the opposite face to β\beta. Next, evaluating α\alpha shows it is adjacent to β,δ,ζ\beta, \delta, \zeta, and ϵ\epsilon (since β\beta and ϵ\epsilon are opposites), leaving γ\gamma as the opposite face to α\alpha. By the process of elimination, the remaining two faces, δ\delta and ζ\zeta, must be opposite to each other.

Step-by-Step Solution

1
Identify all visible adjacent faces for β\beta across the given positions.
In Position 1, β\beta is adjacent to α\alpha and δ\delta. In Position 2, β\beta is adjacent to γ\gamma and δ\delta. In Position 3, β\beta is adjacent to α\alpha and ζ\zeta. Combining these, β\beta is adjacent to α,γ,δ\alpha, \gamma, \delta, and ζ\zeta.
Since a single face of a die can have exactly four adjacent faces, finding all four uniquely determines the fifth face as its opposite.
2
Determine the face opposite to β\beta.
Because α,γ,δ\alpha, \gamma, \delta, and ζ\zeta are adjacent to β\beta, the only remaining letter, ϵ\epsilon, must be opposite to β\beta.
By the process of elimination among the six faces, the one face not adjacent to a given face must lie on the opposite side.
3
Determine the faces adjacent and opposite to α\alpha.
From Position 1 and 3, α\alpha is adjacent to β,δ\beta, \delta, and ζ\zeta. Because β\beta is opposite ϵ\epsilon, α\alpha must also be adjacent to ϵ\epsilon. This means α\alpha is adjacent to β,δ,ϵ\beta, \delta, \epsilon, and ζ\zeta, leaving γ\gamma as the face opposite to α\alpha.
Repeating the deduction process systematically for another face eliminates a second pair of opposite faces.
4
Identify the final opposite pair.
Since we have established that β\beta is opposite ϵ\epsilon and α\alpha is opposite γ\gamma, the remaining two faces, δ\delta and ζ\zeta, must be opposite each other.
Once two pairs of opposite faces on a six-sided die are found, the last two remaining faces form the final opposite pair.

Key Concept

Identifying opposite faces on a 3D die by mapping adjacent faces from multiple rotational views.
Estimated Time:1m 30s
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Topics

Cubes and Dice

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