# Melinda's physical 2x2x2x2 puzzle

#### Marc Ringuette

##### Member
For quite a while, people have been searching for a physical 3D twistypuzzle analog of the 4D Rubik's hypercube. We've found one!

Melinda Green, one of the authors of the java program Magic Cube 4D, has developed a physical analog of the 2x2x2x2 hypercube puzzle that is quite nice, and we'd like to invite you to help us analyze it.

Here's Melinda's intro video, with some useful links in the description.

There's a link to the MC4D mailing list, where the discussion has been happening so far, and a link on how to obtain a puzzle, either by ordering the Shapeways print and some magnets yourself, or by having one of us assemble one for you (at slightly less than our cost, currently $99 US and$125 international). Currently about a dozen people have these; we'll keep mailing them out until we get really tired and have to stop, and then you'll have no choice but to assemble your own.

Let's have some fun figuring out the puzzle.

Some more description of the puzzle, for the people who prefer text to video.

Melinda's 2x2x2x2 is a magnetic twisty-stacky puzzle made from 3D printed pieces. It has 16 four-color cubical pieces, with tetrahedral symmetry of each piece, and 64 tetrahedral "stickers" in 8 colors (four sets of two opposite colors, corresponding to the four 4D axes). The magnets are set up to allow 180 degree (but not 90 degree) twists of individual pieces, although larger 2x2x1 and 2x2x2 chunks of the puzzle can be twisted by 90 degrees and still satisfy the magnetic constraints. The 24 natural orientations of a cube in 3D are divided into normal and inverted or "inside out" versions of the 12 orientations of a 4-color 4D hypercube corner. Yes, this is a bit hard to wrap your head around; that's half the fun.

By defining a suitable set of legal moves for the puzzle, we can emulate the state space of the 2x2x2x2 hypercube puzzle precisely, using the MC4D computer program as our reference for correct behavior of the 4D puzzle. By following along keeping the virtual and physical puzzles in sync, we can satisfy ourselves that we can fully scramble the 4D puzzle, solve it in the 3D analog, and follow along mechanically on the virtual puzzle until it reaches a solved state simultaneously.

Twistypuzzle purists may be a bit disappointed that the puzzle uses magnets, and that there are some moves that are physically possible but violate a parity of the hypercube puzzle, so must forbidden if we wish to keep the puzzles equivalent. It's also a bit disappointing that the solution lengths can differ by a constant factor, depending on the correspondence being used (we can define a 1-1 correspondence of states easily, but a 1-1 correspondence of legal moves is only possible if at least one multi-move "macro sequence" is defined as a "move" in either the physical puzzle, the virtual puzzle, or both). But hey, let's not be complainers: I think we'll learn a lot from it, and besides, it's just frickin' cool.

-- Marc Ringuette

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#### bobthegiraffemonkey

##### Member
I've already done the sim BLD, so I suppose so . Will actually be interesting as I used 3-style before but I expect 3OP to be a better strategy on the physical cube. Might take a while, and I've not got around to getting a copy yet but about to fix that.

#### Scigatt

##### Member
Any thought on getting a company to put it into mass production?

#### Gws

##### Member
Any thought on getting a company to put it into mass production?
I hope, you can get it on shape ways, but it is expensive

#### Marc Ringuette

##### Member
Hi gang! Here's an update.

Over the past year, about 50 people have obtained Melinda's 2x2x2x2, and there are currently 11 entries in the Hall of Fame: people who have posted full solutions, mostly as YouTube videos. Solutions seem to be around 100-300 moves long, and our methods are still a bit primitive. That'll get better soon; some of us on the 4D_Cubing mailing list have been trying to put together some software solvers for the puzzle (an optimal one to find short algorithms, and a 3-phase suboptimal solver for any position). Those software aids should help us a lot in developing better solution methods.

Melinda's main webpage for the puzzle: http://superliminal.com/cube/2x2x2x2/
Melinda's YouTube playlist on the 2x2x2x2: https://www.youtube.com/playlist?list=PLx1mIVtz33hLPgMQ_5DRG1kq1P5GNeKKC
The 4D_Cubing mailing list: https://groups.yahoo.com/neo/groups/4D_Cubing/info

Most of the 50 existing puzzles were fully assembled by me or Melinda and mailed out for varying prices in the vicinity of \$110. If you want one, see the slightly complicated ordering instructions buried down in Melinda's webpage. You email Melinda to get a price and then to get in her build queue; waiting time has been between 3 days and 4 weeks depending if there are any on hand. A handful of people have assembled their own as well, but since we're sending out the assembled puzzles at slightly below cost, almost everybody has sensibly taken the easy route.

Melinda has been sniffing around on how to get them produced more cheaply, but no options have appeared so far.

Someday, we really should put together a Javascript based "virtual physical 2x2x2x2" that emulates the physical puzzle in a browser. It'll be a pale imitation of holding something in your hand, but might make a good gateway drug.

I recently came up with a fairly nice definition for a mini-puzzle using half of Melinda's 2x2x2x2, which is fun and pretty easy, and is also good practice for the larger puzzle because it still has all 12 of the orientations of the corner pieces of a 4D twisty puzzle. I call it the "twisty stacky 2^3" or alternatively the "2x2x2x1". This playlist of mine has two short intro videos, and then a longer one showing an emulation of the mini-puzzle inside the MC4D 2^4 hypercube puzzle.