GoKawiilThis looks like a Rubik’s Cube, but it’s not:
It seems that an open-cab truck filled to the brim with Rubik’s Cubes hit a bump in the road, causing some of the cubes to spill out of the truck bed and onto the roadway. When, mere moments later, those cubes got run over by a magic school bus, something unexpected happened: the plastic pieces of each flattened cube stuck together to form a puzzle of a different kind—one that only looks three-dimensional.
I call it the Flat Cube. Its pieces are rhombuses, also called lozenges, and wherever three lozenges come together to form a little hexagon, you can twist the three lozenges as a unit around the center of that hexagon by any multiple of 60 degrees, like this:
The challenge is to restore a scrambled Flat Cube to its unscrambled flat state using as few twists as possible. And that leads to the question: if the Devil gets to scramble the puzzle as maliciously as possible, how many twists will God need to unscramble it?
Here we are to assume that God, though omnipotent, is scrupulous about following the puzzle’s rules. We’ll also grant both God and the Devil unlimited computational resources; God can always find the most direct solution to any scramble the Devil devises, while the Devil wants to make that solution involve as many moves as possible.
In the case of Rubik’s Cube, this magic number—the number of moves God needs against the Devil’s most devilish scramble—has been dubbed God’s number. God’s number is known to be 20 if a 180-degree twist counts as one move and to be 26 if a 180-degree twist counts as two moves.
In the case of the Flat Cube, I don’t know God’s number. But whether you count twists in 60-degree increments or allow a twist by any angle to count as one move, I can show you that God’s number for the Flat Cube is at least 27.
There are two wonderfully visual ways to see why this is so; one works by adding a dimension, while the other works by taking a dimension away.
ADDING A DIMENSION
When you look closely at the Flat Cube, you may begin to experience dizziness. Lean into this feeling. It’s trying to help you.
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