# Melinda's 2x2x2x2

This puzzle is a true 4D analog of the 2x2x2 Rubik's cube. I believe it is the world's first physical 3D embodiment of a 4D twisty puzzle. Specifically, the 24 The video above shows how it works, what the legal moves are, and the basic information you need to use it. A short follow-up video lists the set of canonical moves. For the mathematically curious, Marc Ringuette made a wonderful video showing the correspondence between the physical and virtual puzzles. The main discussion group for all higher-dimensional puzzling is the hypercubing Google group. There is also an active Hypercubers Discord Server where speedsolving algorithms and related puzzles are being developed. Feel free to join the discussions.

The number of possible states for the 4D cube is exactly

16!×12^16/(6×192)

or in decimal as
3 357 894 533 384 932 272 635 904 000

This means the 24 has about a billion times as many states than Rubik's original 33.

Ernő Rubik examines the puzzle

Above is professor Rubik himself examining one of my puzzles at the 13th Gathering For Gardner conference. It was reported that his only comment was that none of the derivative puzzles matter and that only his original invention is important.

## History

My friend Don Hatch and I came up with the idea for a virtual 4D Rubik's cube almost 30 years ago. We then wrote the first version of MagicCube4D and have been porting it from platform to platform ever since.

Almost from the beginning, we and others in the community that grew around it wondered if one could ever make a physical version of it. It's natural to want to reach in to touch it and operate it directly, but the physical requirements made a physical version seem incredibly unlikely. Still, I could never completely stop thinking about that. Deciding to focus on a 24 rather than the full 34 was helpful, but still there was no clear way to achieve even that.

In early 2014, during a discussion with Oskar van Deventer, I sent him some rough sketches of the pieces and topology involved, which he turned that into a beautiful rendering, although neither of us had any idea for a mechanism that would allow it to function. The rendering was very inspirational, however, and I kept coming back to it. Eventually I had the idea of stretching it into a less symmetrical configuration and squashing the pieces into cubes.

At that point, I seemed to have a design for a potentially workable puzzle but still had no idea for a mechanism. I figured that magnets were probably my only hope, but how to do that was far from clear. Around the end of 2016, I stumbled onto a Mathologer video about magnetic Rubik's cubes made from dice. It included a magnetic arrangement that allowed for a workable 23, and I realized it might be extended to do what I needed. I built my first prototype and thought I may have accomplished it.

Then we discovered that it wasn't exercising the full state space of the 4D puzzle, and that was a big setback. Eventually I realized that I could reach the full state space if I could just find a kind of 4D rotation that would swap the outer axis with any of the other three. Eventually, I found such a way and reduced it to a short enough sequence to be practical. We came to call this a "gyro" move.

Then there was a final setback when I found that the magnetic arrangement wasn't quite general enough to support such a transformation. But Matthew Sheerin quickly realized that that could be fixed at the expense of doubling the number of magnets to 384. That's a lot of magnets, but who cares about that if it works!

This was when I needed to move to 3D printing, so there was a lot of learning and experimenting to make a printable design at any kind of reasonable price. It was somewhat affordable, but eventually, the cost of SLS printing rose to the point that few could afford it. I then needed to turn to injection molding to cut the price considerably. It took 3 years of R&D, but May 2022, I finally succeeded, and the result is far better than I hoped and at a fraction of the cost. I still think it's a kind of miracle that this all came together after so very long.

The success of this puzzle immediately begged the original question of whether the holy grail of a physical 3x3x3x3 might be achievable with my general design. I still didn't see how, but Grant and Luna on the Discord Server had the brilliant idea to focus first on intermediate 4D cuboids. They quickly came up with a design for a workable 2x2x2x3 and soon had designs for all the cuboids up to and including the 3x3x3x3! Grant then began the laborious process of printing many hundreds of pieces and connectors, and the even more laborious task of assembling them all. See and learn about these amazing puzzles in the Superliminal wiki here.

As of 2024/6/5