Roll the Cube Riddle: Which Symbol Is on Top at the End?
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The triangle is on top at the end. The cube makes nine rolls, and if you track the top face after each one, the last roll brings the triangle up.
How do you follow the cube roll by roll?
At the start the circle is on top, the diamond is underneath, the star faces the back, the square faces the front, the hexagon faces left and the triangle faces right. When the cube tips over an edge, the face on the side it rolls away from comes up to the top.
| Roll | Direction | Top face after the roll |
|---|---|---|
| 1 | Right | Hexagon |
| 2 | Right | Diamond |
| 3 | Up the page | Square |
| 4 | Up the page | Circle |
| 5 | Right | Triangle |
| 6 | Right | Diamond |
| 7 | Right | Hexagon |
| 8 | Down the page | Square |
| 9 | Down the page | Triangle |
Is there a shortcut?
Yes. Two rolls in the same direction swap the top and the bottom, and four rolls in the same direction bring the cube back to where it started. So you only need to think hard at the corners, where the direction changes.
A common slip is to miss one of the three rolls along the top row, which leaves the circle on top instead of the triangle.
Why do cube puzzles reward careful tracking?
Small changes to a cube add up fast. With six colours, one per face, a cube can be painted in 30 different ways that cannot be turned into each other [1]. Puzzles solved one move at a time can also run long: the sliding 15-puzzle can need as many as 80 moves [2].
Ready to roll again?
These riddles all ask you to move or fold a shape in your head.
References
- Siegrist, R. (2018, September 25). 2 × 2 × 2 Color Cubes. Recreational Mathematics Magazine, 5(9), 65-70. DOI 10.2478/rmm-2018-0003
- Parberry, I. (2015, July 10). Solving the (n² − 1)-Puzzle with 8/3 n³ Expected Moves. Algorithms, 8(3), 459-465. DOI 10.3390/a8030459