Examkin

Question

Difficulty: MediumCubes and Dice

A standard six-sided die (where the sum of the numbers on any pair of opposite faces is exactly 77) is placed on a flat table. In its initial position, the number 44 is on the top face, the number 22 is on the front face (facing you), and the number 11 is on the right face.

The die is then rolled 90∘90^\circ over its edges along the table in the following sequence:
1. Rolled to the Right (East).
2. Rolled Forward (North, away from you).
3. Rolled to the Right (East).
4. Rolled Backward (South, towards you).

What number will be on the top face of the die in its final position?

  1. 6Answer
  2. B
    3
  3. C
    5
  4. D
    1

Answer

The number 6 will be on the top face.
By carefully tracking the 3D rotation of the die step-by-step, the top face transitions from 4 to 6 (after the first right roll), to 2 (after the forward roll), to 3 (after the second right roll), and finally to 6 (after the backward roll). The sequence of orthogonal rotations guarantees that the number 6 rests on the top face at the end.

Step-by-Step Solution

1
Determine the initial state of all six faces based on standard die rules.
Top = 4, Bottom = 3, Front = 2, Back = 5, Right = 1, Left = 6.
A standard die has opposite faces that sum to 7. This allows us to deduce the hidden faces from the visible ones.
2
Track face positions after the first roll: 90° to the Right (East).
Top becomes 6, Bottom becomes 1, Front remains 2, Back remains 5, Right becomes 4, Left becomes 3.
Rolling Right pivots the die over the right edge; the Left face (6) moves to the Top, while Front and Back remain unchanged.
3
Track face positions after the second roll: 90° Forward (North).
Top becomes 2, Bottom becomes 5, Front becomes 1, Back becomes 6, Right remains 4, Left remains 3.
Rolling Forward pivots over the back edge; the Front face (2) moves to the Top, while Right and Left remain unchanged.
4
Track face positions after the third roll: 90° to the Right (East).
Top becomes 3, Bottom becomes 4, Front remains 1, Back remains 6, Right becomes 2, Left becomes 5.
Rolling Right again moves the new Left face (3) to the Top, while Front and Back remain unchanged.
5
Track face positions after the final roll: 90° Backward (South).
Top becomes 6, Bottom becomes 1, Front becomes 3, Back becomes 4, Right remains 2, Left remains 5.
Rolling Backward pivots over the front edge; the Back face (6) moves to the Top. Thus, the final number on the top face is 6.

Key Concept

Spatial tracking of rolling dice and orientation of opposite faces.
Estimated Time:1m 30s
Rate this question

Topics

Cubes and Dice

More questions from this topic