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Closed loop on a side length 2 cube Question
Place 4 tiles on each face of a cube so that the arcs on all 24 tiles form a single closed loop.
If you want to try a simpler version first, there is Closed loop on a side length 1 cube.
You may rotate each tile by any multiple of 90 degrees.
The tiles to be used are equal numbers of 2 kinds of tile:
- 12 single arc tiles:
- 12 double arc tiles:
Here are 12 of each that can be printed and cut out:
Here is a net of a cube to place the tiles on:
Each of the images can be opened in a separate tab (which will display the tiles and the squares on the cube faces at exactly the same size) if you would like to print them for experimenting or to photograph your solution to post in an answer.
Hint
This is not intended as a brute force puzzle where you try all the combinations until you find one that works. There are some insights that will make this puzzle practical to solve even for larger versions.
1 answer
The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| trichoplax | (no comment) | Sep 14, 2026 at 17:10 |
Solution
Process
I start with a fairly symmetric placement which doesn't meet the constraints and make simple modifications until it does.We have two "single" and two "double" tiles per side. We can place them to leave four vertices of the cube clear:
Now place the double tiles in the remaining spaces:
The only problem is that we have more than one loop. For each double tile in some arbitrary order, if rotating it 90 degrees will join two loops then do so.
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