Message #3641

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
Date: Fri, 10 Feb 2017 21:15:50 -0800

Hello Matt,

First off, I’m thrilled that you think the problem is solvable! I like your focus on making sure all piece orientations are possible. Your first solution looks like the right approach. No worries for not understanding the problem until now. It is one of the important reasons to prototype. Yes, 384 magnets is a lot but it’s only double what I’m currently using. It may drive the material costs up to $100 but puzzle enthusiasts routinely pay more than that for special puzzles. It does add a scary dimension to the thought of a 3^4 version, but first things first.

Your second solution using free-turning magnetic balls is very clever. I suspect it would only be practical if someone already makes such a unit. It doesn’t need to be spheres though. It could be ordinary bar magnets on spindles. Yet another option occurred to me which is that not all of the embedded parts need to be magnets. Some could be ordinary unmagnetized ferromagnetic material since it will be attracted to both polarities. I don’t see how to use the idea yet but it’s another option.

Regarding moves allowing access to the full state space, I don’t fully understand your suggestion. It sounds like you are suggesting a restacking move where one 2x2x1 block is moved from one end to the other but not flipped, is that right? EG starting with the two halves of the gray face visible on the ends, all the gray stickers will end up physically touching, just like the black face in the center. The problem is that leaves the puzzle in one of those strange inverted cases and doesn’t seem to move stickers off of the red-blue faces.

One move that it does seem to allow is rotating both end caps 90 degrees onto opposite sides of the central 2x2x2. In other words, a 90 degree twist of the central (black) face but in a direction that crucially does move stickers off the red-blue faces. Double moves like this are slightly inelegant, but that’s a small price to pay for a complete solution. One nice thing about this twist is that it gives meaning and purpose to the interior physical stickers. Currently, half of the physical stickers are not really needed because there are no ordinary twists that leave them exposed. My current bandaged puzzle would even work better without them.

So thanks for the great ideas. I’m going to see about ordering some pieces with your new configuration.

Now here’s a mini-puzzle: Can anybody describe a sequence of moves on this puzzle that results in reorienting a single piece? Bonus points for short sequences that use the fewest double moves. If there’s a short enough sequence, I may show it in the video. One ironic thing is that the fact that you can always do it, means that it’s valid to simply yank out a single piece and reorient it directly! Is that cheating or just another shortcut? Interesting stuff.

Happy puzzling!
-Melinda

On 2/10/2017 3:52 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
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> Melinda,
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> It seems that the choice of magnet arrangement isn’t correct then. Fortunately, I think that’s solvable. I now wish I’d fully understood this approach when you first discussed it, it took this prototype and some careful thinking before I finally clicked.
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> First of all, I think that taking a 2x2x1 block from one end and putting at the other end is a valid maneuver, a 4D rotation which will allow blue/red pieces to mix with other colours and to fully scramble the 2^4.
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> Then, how should the magnets be configured? Well, I see two possible solutions.
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> 1) We need to be able to pick out any piece and put it back in any of the 12 orientations of 2^4 corners (which are the rotational symmetries of the tetrahedron) and be able to attach it back to the puzzle, which tells us the symmetry of any valid static arrangement of magnets. I skeched a solution quickly and uploaded it to the group (it’s in the group files in my folder since I’ve no idea how to attach images to a post): shown are 3 sides of one piece and 12 magnets embedded in those faces, with the colour of the dot indicating the orientation of the magnet. As a quick desciption, there are 4 magnets on each face in a ‘+’ pattern, with orientation of the magnets alternating as you go round the face. This is done such that if the cube is rotated about a corner the configuration looks the same. I believe this will result in a fully-functioning puzzle, but it takes 384 magnets!
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> 2) It might be possible to have a spherical magnet in a cavity in the center of each of the 6 faces of each piece. When two faces are brought close to each other, the magnets should be able to realign so that they attract. This only takes 96 magnets, but I’m less convinced it will work well and robustly than a static configuration of magnets.
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> If I’ve made some mistake or someone has a better idea, I’d be interested in seeing it. Either way, I hope a physical 2 ^4 is available to buy some day, it would be so much fun to play with and show off!
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> Matt
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