Hi, I'm Henry Segerman,
and this is the Helix cube puzzle.
It sounds a bit like the Rubik's cube puzzle, but this is quite different.
It's a cube, it has lots of little coloured "tiles", if you like.
There's one missing tile,
which means that you can move one of the neighbouring tiles into the hole.
This then leaves room that another tile can move into, and so on.
To better see what's going on, let's look at this prototype of the frame.
This is only half of the cube, so you can see the inside.
Each edge of the frame
has two helical fins, that twist slightly from one end of the edge to the other.
The tiles have slots cut into them that fit onto the helical fins.
As you slide the tile along one of the edges, it follows the twist of the fins.
When you get to the end of the edge, the tile is rotated by 90
degrees, doing a right handed screw motion along the edge.
This is the same basic puzzle idea as the 15 puzzle.
You mess up the tiles
and then you try to get them back to the correct places again.
I've made a few different variants of the 15 puzzle,
starting with the 15 + 4 puzzle.
This is the same as the 15 puzzle, but with four extra tiles.
More importantly, this puzzle also adds "holonomy".
That is, when you move a tile around in a loop, it might come back rotated.
Here you can see that the 11 tile is the same way up as the 10,
which is the same way up as the 6 which is the same way up as the 7,
which is the same way up as the 16.
But now if you look at the 11 and the 16, they are rotated by 90 degrees.
So if I move the 11 tile all the way around in the same way,
it would come back rotated by 90 degrees as well.
So to solve the puzzle, you not only need to get the tiles
back in the right places, you also need the rotations to be correct.
Another way you could alter
the 15 puzzle is to go from two dimensions to three dimensions.
This is the Rubik's slide puzzle.
Again, there are tiles with one tile missing so the others can slide around.
In the 15 puzzle, the tiles slide around a square four by four grid.
In the Rubik's slide puzzle, the tiles slide around the edges of a cube
with three tiles fitting along each edge.
The Helix cube puzzle sits over here.
It's what happens when you add holonomy to the Rubik's slide puzzle.
Having said that, for that to be
holonomy, tiles need to get rotated when we go around a loop.
Well, do they?
Let's take a look.
And yes, I'm very deliberately doing this with a blank tile
so as to not spoil this puzzle if you want to think about.
When we go around a square doing a right handed
screw turn on each edge, do you end up rotated at the end?
And if we do get rotated, around which axis and by what angle?
Don't worry, I'll show you the answer at the end of the video.
This is not at all easy to visualize.
But first, as with all of the holonomy puzzles,
the physical design has to stop you from rotating the tiles in place.
All of the rotation must come from the pieces moving around the frame.
When you're in the middle of an edge, the fins stop
you from doing anything other than sliding along the edge in either direction.
When you're at a corner it's trickier because you need to be able to move
in three different directions and still not be able to rotate.
That's what these grooves on the outside of the tile are for.
They fit onto tripod shapes on the frame
and don't allow rotation along any axis as long as the tile is in the corner.
If you try to move away from the tripod, the only way to do it
is to move along one of the three edges, at which point
you're prevented from rotating by the fins.
This is a good place to talk about an open problem here.
Suppose you wanted to make a bigger frame,
perhaps with two of these cubes stacked next to each other.
You wouldn't be able to have a full tripod at the corners where the cubes meet.
This leg of the tripod would stop you from moving a tile further this way.
So I think you might not be able to prevent rotation in place.
Or maybe it would just barely work.
But if you had a corner where eight of these cubes met,
then you couldn't have anything like the tripod to block rotation.
So is there some other design that could stop rotation in place instead?
Ok, back to the full helix cube puzzle.
Thanks to Sabetta Matsumoto for coloring the faces of the tiles
with sharpie markers.
We thought that we might be able to mask off sides of the tiles
and use spray paint, but the test did not go well.
Coloured stickers wouldn't work very well either because the shapes of the tiles
are so complicated and full colour 3D printing would have been very expensive.
So this is the best we could come up with.
The helical edges on the full
puzzle are thicker than those on the half cube prototype.
The reason is that these three edges have bolts inside of them.
If we undo the bolts,
this corner comes off,
which is how you can get the tiles into the puzzle in the first place.
I could have had
just these three edges be thick and not thickened up all of the rest of them.
But I think they would have looked a bit unbalanced,
so I did all of them.
Ok, as promised, here is the solution to the puzzle from before.
Here's what this tile looks like at the start.
And now I'm going to send it around the square doing a right
handed screw turn every time I move along an edge.
And when I get back to the start, it looks like this.
How are these related to each other?
These are related by a 120 degree rotation
around a long diagonal of the cube.
All right. That's the Helix cube puzzle.
If you've been following my explorations of these variants of the 15 puzzle,
maybe the next question that comes to mind is "Ok,
can you get from any state of this puzzle to any other state
by moving the tiles around?" Or are there some states that are inaccessible -
that you cannot get to?
The set of all possible
configurations of a puzzle breaks up into connected components.
If there are inaccessible states, then there's more than one component.
And then you can ask "How many components are there?"
I'll leave you with the very satisfying, most efficient
way to pack the tiles into a cage for printing.
And this is how I got those tiles out
of the cage.