This puzzle takes place on the surface of the following gridded, beveled cube:
The surface of this cube is divided into 3458 small regions separated by black lines. Of these regions, 3450 of them are squares, and 8 of them are triangles. Of the squares, 3174 are on the flat faces of the cube (23 × 23 = 529 per face, × 6 faces = 3174), and 276 of them are on the beveled edges (23 per edge, × 12 edges = 276). The 8 triangles are at the corners.
If you look very closely, you will notice that this cube has an ant crawling on it. This is no ordinary ant. It is a magical, immortal ant born* to pursue a single goal. The ant's goal is to find a single, continuous path that will take the ant through all of the 3458 regions on the cube, each exactly once. The ant has been trying to find a way to do this for a long, long time, and has so far never succeded.
Can it be done? Your task is to either prove this is possible, or prove it is not.
Some rules, caveats, and clarifications:
- The path must visit each of the 3458 regions (all 3450 squares, and all 8 triangles) exactly once.
- The path may start in any region, and may end in any region.
- The ant may only move between regions that share a border. (The regions can't just touch at the corners.)
- Assume that the cube is magically suspended in the air, and that the ant can walk equally well on any surface in any orientation without risk of falling off.
- The ant never gets tired, hungry, bored, will never die of old age, be eaten by a lizard etc., etc. This is not a lateral thinking puzzle. The solution is based on a simple mathematical insight.
* OK, fine. Hatched to pursue a single goal.