Comment by Oreb | Hacker News Reader

Comment by Oreb | Hacker News Reader

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From a puzzle-solving point of view, these very large cubes aren't that interesting. When you increase the cube size, there are new things to figure out, but only up to a certain point. Figuring out how to solve a 4x4x4 when you know how to solve a 3x3x3 takes some significant work. I think I spent a whole weekend to successfully solve a 4x4x4 the first time I got one, despite being reasonably good at solving the 3x3x3. Solving a 5x5x5 for the first time took just a couple of hours, there wasn't much new to learn. The 6x6x6 was easier still. When I got to the 7x7x7, there wasn't really anything new at all. I could solve it immediately, it just took more time.

Anything beyond 7x7x7 is pretty much the same. It's just more annoying, because the puzzle gets physically harder to handle, and because you have to do the tedious work of counting how many layers away from the centre a piece is. The 7x7x7 is the biggest cube used in official competitions, for a good reason.

The motivation for making enormous cubes like the 34x34x34 is just the engineering challenge, and breaking records. Nobody is going to want to solve such a thing, at least not more than once.

That all sounds like fun but I'm still working through solving a 64-disc Tower of Hanoi puzzle right now and won't be able to get to another puzzle for a bit.

I'm still waiting for my Moment of Glory when a puzzle room or something has a Hanoi tower and I can slam out the solution as quickly as I can move the pieces, thus justifying all my formal Computer Science education once and for all.

(There is a very easy-to-remember algorithm that can be trivially executed by humans given here in a Mathologer video, with a time-code link to jump straight to it: https://youtu.be/MbonokcLbNo?si=ey8bv4T9KbDxgB7N&t=650 )

If the Hanoi pieces alternate in color, there is another very easy algorithm: always avoid putting two pieces of the same color directly on top of each other. I noticed this by accident while watching someone else solve a Hanoi puzzle with alternating colored pieces. (The person I was watching didn't know this trick either. I suspect the manufacturers of the alternatingly colored puzzle also didn't know about it. So where did the knowledge come from?)

Lol, minimum moves needed 2^64-1

Just to elaborate: Solving a 5x5x5 or a 7x7x7 is basically just turning the cube into a 3x3x3 by lining up the edges and fill in the centers. Which is a new thing, but quite easy to figure out. And then solve it as if it was a 3x3x3.

That's not the only way to solve big cubes, but it's indeed the most common way (known as "reduction"), and what most people naturally come up with if they try to solve 4x4x4 or bigger on their own. In addition to what you said, there is also the issue of parity (basically, when you reduce a 4x4x4 to a 3x3x3 by solving centers and edges first, you will often end up with a 3x3x3 cube in an unsolvable state, and you need to figure out some tricks to convert it to a solvable state), but if you know how to solve parity problems on a 4x4x4, you can do it for a cube of any size.

Just out of curiosity (no rubiks cube affinity at all), but how can there be an unsolvable state when there are 'tricks' get in a solvable state? Does that not imply that there are no unsolvable states at all? Or is that maybe related to a certain method of solving?

The reduction method means reducing a big cube (NxNxN for N>3) to a 3x3x3 cube by first solving the centers (the central (N-2)x(N-2)x(N-2) square on each face) and the edges (the inner N-2 pieces along each edge of the cube). You are then essentially left with a 3x3x3 cube that you can try to solve by only turning the outer layers (which won't break the centers and edges you solved in the first stage).

The problem with this is that you may end up with a 3x3x3 cube that is not solvable. For instance, you can get a state where the entire cube is solved, except for two edges that need to swap locations. This isn't possible. In group theoretical language, only even permutations are possible. You can swap two _pairs_ of edges, but not just two edges.

When you end up in such an unsolvable 3x3x3 cube, you have to temporarily turn the inner layers of the cube and break apart the centers and edges you built in the first step, and then reassemble them again to a solvable 3x3x3 cube.

thanks!

They mean that the outer 3x3 is unsolvable taken in isolaton. The tricks will involve unsolving the middle faces and solving them again.

thanks!

The moves to fix parity made the 4x4x4 less fun for me. The recommended solution is long.

The hollow 3 has a similar problem. Because you can’t see the central piece there’s a way to rotate the core and a couple of edge pieces so they look like they violate parity.

i was clicking around on the site and found an interesting article about other attempts to make cubes more challenging --

https://ruwix.com/twisty-puzzles/bandaged-cube-puzzles/

in particular, "bandaged cubes" in which certain faces have fused blocks to limit your available moves, and "constrained cubes" in which certain faces can only rotate in one direction, and only by a certain amount.

One of the hardest Rubik's cube I have seen is a regular 3x3x3, but with stickers that change color depending on the angle you look at them from.

I'm not a cuber or a puzzle guy or a math guy, but I am curious: how do you know when it's solved? Or is this a 'whoosh' moment and I'm missing the obvious?

when it's solved you look at all blocks on a face from the same angle, so they all have the same color. the problem is, while you are solving you don't know from which angle you need look at one block. you see a red one on one face, and another red one on another, and when you bring them together you realize that you looked at one of them from the wrong angle and they don't actually match. so effectively you need more moves until you find the right block.

I'd like to try that! Do you remember what it was called?

There is the "Rubik's Impossible"

https://www.rubiks.com/products/rubiks-impossible

Sticker sets are also available, like this one

https://oliverstickers.com/two-face-3x3x3.html

The engineering challenge of making such a 34^3 cube is way higher than that of solving. It's incredible impressive what dedication is capable of.

Thanks for explaining. I assumed this was the case, but want sure. Even the photo of the guy with it annoyed me. Just looked obnoxious, fake, and self-promoting. The guy's YouTube channel also appears to target audience aged 12-14, with obnoxious, juvenile thumbnails to match. Awful.

It is my understanding that a 5x5x5 is actually more similar to a 3x3x3 than a 4x4x4 is.

Sort of. The 3x3x3 and 5x5x5 both have fixed, immovable centers. Red is always opposite orange, blue is always opposite green, and yellow is always opposite white. The 4x4x4 doesn't have fixed centers. When you build the central 2x2 squares on each side (the first step of the reduction method), you have to be careful to have the colors arranged in the correct locations relative to each other. In a certain sense, this is trivial, but it forces you to remember exactly where all colors are on a solved cube in order to solve a 4x4x4 (or other even sized cubes). Odd sized cubes don't have this problem.

Another annoying thing about 4x4x4 compared to 5x5x5 is that you have two possible types of parity issues on the 4x4x4. On the 5x5x5, only one of these can occur.

Nevertheless, if you know how to solve a 3x3x3 and no bigger cube, a 4x4x4 is certainly the easiest next step.

Remember where the colours are when solved is overstating it a bit, you can look at the corners for the answer. Otherwise yep.

Interesting, make me wonder what are the well known algorithms to solve them and how they compare in term of complexity.

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