#### 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

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

Last edited by a moderator: Jan 10, 2018