Rubik's Cube of DOOOOM!!!

[Absol197]

So, I've started a new Frostburn campaign with my IRL friends, and one of the characters' backstories involves a nifty little artifact. It's function is unimportant to the discussion of this thread (unless you'd like to know, then just ask!), but it's construction is:

First, imagine one of those Chinese puzzle boxes, with the little sliding tiles (one square is always empty, allowing the tiles to slide into other positions), 7x7. Each tile is marked with parts of arcane runes, so different arrangements form different runes.

Next, imagine that each tile can be popped up above the others, rotated into a different orientation, then dropped back down to rejoin the puzzle (essentially giving each tile 4 possible orientations).

Now, say that there are 6 of these puzzles on each side of a cube.

Finally, imagine that this cube can be rotated like a Rubik's cube, except it has six rows/columns, and there are six different diagonals that it can rotate on, too (each of the tiles can be broken up along all these different lines, and "snap" into place once the turn is completed so every side always has the same number of complete tiles).

So, ignoring the impossibilities in this thing's construction (magical smiths with incredible crafting ability were used, plus it's magic :smalltongue: ), is it possible to calculate how many different combinations of tile positions are possible on this box? I know it can be done for the puzzles on each side, even taking them in together, but I think once we get rotating the sides into it it's impossible to calculate...

Thanks for any help you can provide! I'm bad at maths, but I wanted to get an idea of how many possible positions there are.


~Phoenix~

[Chronos]

I'm pretty sure that that gives you enough freedom to move any tile anywhere you want, with no conserved symmetries. This simplifies the calculations, but it also means that the numbers get really huge, really quickly. If one ignores the diagonal Rubiking, we've got ((6*49) choose 6)*(6*48)!*4^(6*48) possibilities. According to Wolfram (http://www.wolframalpha.com/input/?i=%28%286*49%29+choose+6%29*%286*48%29!*4^%286*48 %29), this is approximately 1.5*10^770. Meaning, you'd need 771 digits just to write out this monster of a number.

[Absol197]

I'm pretty sure that that gives you enough freedom to move any tile anywhere you want, with no conserved symmetries. This simplifies the calculations, but it also means that the numbers get really huge, really quickly. If one ignores the diagonal Rubiking, we've got ((6*49) choose 6)*(6*48)!*4^(6*48) possibilities. According to Wolfram (http://www.wolframalpha.com/input/?i=%28%286*49%29+choose+6%29*%286*48%29!*4^%286*48 %29), this is approximately 1.5*10^770. Meaning, you'd need 771 digits just to write out this monster of a number.

:smalleek: And that's before taking into account the diagonal turns?

Wow...Good luck trying to solve that! Even if you had a picture of how it's suppose to look, a normal person might never be able to solve it...

EDIT: What's the "choose 6" about? What do each of those factor represent? I'm trying to get it straight in my head for later...

[Rubik]

My avatar resents that!

[Absol197]

My avatar resents that!

No offense, but I think you're avatar barely represents a fraction of this box's puzzle-potential :smalltongue: !

[Rubik]

I know this is totally off-topic, but given your sig, I figured you'd enjoy and appreciate this: http://www.fanfiction.net/s/5398503/1/Embers

[Chronos]

EDIT: What's the "choose 6" about? What do each of those factor represent? I'm trying to get it straight in my head for later...
First step: There are six blank tile spots on the puzzle. Figure out where you want to put those blanks. This is the ((6*49) choose 6) factor: There are (6*49) spaces, and you're choosing six of them to be the blank ones.

Second step: Of the (6*48) spaces that have tiles in them, decide which one goes where. This is just a permutation, for the factor of (6*48)! .

Third step: Each of those 6*48 tiles can be rotated four different ways. This gives the factor of 4^(6*48).

Ironically, this puzzle would probably be much easier to solve than a standard Rubik's cube, since the same freedom that allows for all of these possibilities also gives the solvers many options. Consider: If you take a standard Rubik's cube, but you're allowed to peel the stickers off, the number of possible configurations increases significantly... But at the same time, it's also obviously easier to solve. Well, all of the many possible manipulations of this puzzle mean that, effectively, you're able to do the equivalent of peeling the stickers off.

[Absol197]

First step: There are six blank tile spots on the puzzle. Figure out where you want to put those blanks. This is the ((6*49) choose 6) factor: There are (6*49) spaces, and you're choosing six of them to be the blank ones.

Second step: Of the (6*48) spaces that have tiles in them, decide which one goes where. This is just a permutation, for the factor of (6*48)! .

Third step: Each of those 6*48 tiles can be rotated four different ways. This gives the factor of 4^(6*48).

Ironically, this puzzle would probably be much easier to solve than a standard Rubik's cube, since the same freedom that allows for all of these possibilities also gives the solvers many options. Consider: If you take a standard Rubik's cube, but you're allowed to peel the stickers off, the number of possible configurations increases significantly... But at the same time, it's also obviously easier to solve. Well, all of the many possible manipulations of this puzzle mean that, effectively, you're able to do the equivalent of peeling the stickers off.

Really? It doesn't seem like it to me, but I guess I'm not as well versed in these subjects as you are. Because each tile is split up into several pieces, and the "solution" would have each of those individual pieces needing to be in a specific place. The algorithms for switching one piece to where it needs to be without messing with any of the others' positioning would be huge! And especially because no computer would be available to help design that...

I mean, first you would have to create all the correct tiles in the correct orientation, which would take forever. And then you'd have to get them all on the correct sides, but no twist of the cube moves an entire tile at once (well, some of the diagonals might, I think), and then you have to solve the puzzles on each side. Yes, the tiles being able to slide helps (in that you don't have to do a complex series of moves to get a piece from the top of one side to the bottom), but what if that tile has one quarter of itself that is wrong, and the missing quarter is split into three pieces each on different sides of the cube? That would take forever just to get that one tile composed correctly!

[Madcrafter]

I'm wondering how you fit six squares exactly into a larger square? Bending space? And six diagonals?

And at least from my reading of the OP, that math missed a step or two, and the actual number would be much larger.

As I read it:

We have a cube.
Each of the six sides of that cube has six puzzles.
Each puzzle is a 7x7 grid, with on blank square, that form different runes.
These are divided into six rows, and six columns, each with eight(?) degrees of freedom, that can be set to four positions. I'd simplify that to agree with Chronos and say it basically allows you to set the tiles to whichever location you want.

Assuming the contents of the 7x7 boxes can be mixed at will, that gives the 6912 tiles 7056 different possible positions each. It's too late here for combinatorics, but as a simple permutation of 6948 items (adding in the 36 blank spaces), gives 1.213e23678 combinations. A number over twenty-three thousand digits long (over 230 googol!). So yeah, it'd be pretty hard to solve without divination magic.

Of course, the above answer is probably off too, but I think it might be a little closer than the one above. I think you'll have to clarify a little better how the cube is made up.

[unseenmage]

It's the Hellraiser Puzzle Box. As an Artifact. Neat.

[Waker]

It's the Hellraiser Puzzle Box. As an Artifact. Neat.

Curse you unseenmage, I was gonna say that. Well, that's what I get for coming into the topic late.

[Chronos]

I was ignoring the diagonal twists, in part because I'm not really sure what those were meant to describe (how do you put six diagonals on a cube?). If included, those will of course increase the number of possibilities.

Absol197, don't think of it as "first you have to do this part, then you have to do that part, and then you have to do that other part". All of the types of moves are available at all times, which gives you a lot more options, and makes it a lot easier to find an option that does exactly what you want it to.

[Karoht]

So what exactly does the puzzlebox of doom do?

If you solve a side you get X?
If you solve all sides you get something even more awesome than X?

Also, how is the party expected to solve this puzzle or interact with it?

[Absol197]

I'm wondering how you fit six squares exactly into a larger square? Bending space? And six diagonals?

And at least from my reading of the OP, that math missed a step or two, and the actual number would be much larger.

As I read it:

We have a cube.
Each of the six sides of that cube has six puzzles.
Each puzzle is a 7x7 grid, with on blank square, that form different runes.
These are divided into six rows, and six columns, each with eight(?) degrees of freedom, that can be set to four positions. I'd simplify that to agree with Chronos and say it basically allows you to set the tiles to whichever location you want.

Assuming the contents of the 7x7 boxes can be mixed at will, that gives the 6912 tiles 7056 different possible positions each. It's too late here for combinatorics, but as a simple permutation of 6948 items (adding in the 36 blank spaces), gives 1.213e23678 combinations. A number over twenty-three thousand digits long (over 230 googol!). So yeah, it'd be pretty hard to solve without divination magic.

Of course, the above answer is probably off too, but I think it might be a little closer than the one above. I think you'll have to clarify a little better how the cube is made up.

I'm sorry, I guess I didn't describe it well enough. There is only one 7x7 puzzle on each side of the cube. There are six total puzzles, one on each side.


I was ignoring the diagonal twists, in part because I'm not really sure what those were meant to describe (how do you put six diagonals on a cube?). If included, those will of course increase the number of possibilities.

What I was originally thinking was this:

Imagine you have a pizza, and you cut it into 16 equal pieces. To do so you would make eight cuts. If we took out the cuts that were exactly vertical and horizontal, then the six remaining ones would be the diagonals - the two-catty-corner cuts, and four more.

However, thinking this through more (based on your comments) has me thinking that while this might work for a square, I don't think it would really work for a cube. Unless you think it would, I'm going to decrease it down to just two diagonals, the two corner-to-corner ones.


Absol197, don't think of it as "first you have to do this part, then you have to do that part, and then you have to do that other part". All of the types of moves are available at all times, which gives you a lot more options, and makes it a lot easier to find an option that does exactly what you want it to.

True, but like a normal Rubik's cube, every twist of the cube affects every side of the cube. Sure, you can do the twists and turns necessary to get a tile assembled the proper way, but that's going to disassemble several other tiles and move the pieces to various points on the cube. Putting the one tile you're looking for together should be simple enough, but keeping that one together while you assemble all the others? I think that would be rather difficult. But maybe I'm imagining it wrong.


So what exactly does the puzzlebox of doom do?

If you solve a side you get X?
If you solve all sides you get something even more awesome than X?

Also, how is the party expected to solve this puzzle or interact with it?

It controls fear. So once it's assembled (each of the six sides and the main gears in the center), it gains minor powers. Activating it requires putting it in several solutions in a row (seven, to be exact). The more correct solutions its been in in a row, the stronger its powers become, until it's fully activated, at which time the holder can control the fears of anything. Basically, it's spawn effectively living phantasmal killers and living weirds, but as outsiders instead of oozes, and more powerful, plus give a lot of other cool, OP abilities.

The players aren't supposed to use it :smalltongue: ! They're trying to stop it from being used! The bad guys are trying to assemple it, and their capturing midgard dwarves (the race that built it in the first place) to tell them how to put it back together, and hoe to activate it.

[Karoht]

It controls fear. So once it's assembled (each of the six sides and the main gears in the center), it gains minor powers. Activating it requires putting it in several solutions in a row (seven, to be exact). The more correct solutions its been in in a row, the stronger its powers become, until it's fully activated, at which time the holder can control the fears of anything. Basically, it's spawn effectively living phantasmal killers and living weirds, but as outsiders instead of oozes, and more powerful, plus give a lot of other cool, OP abilities.Nifty.


The players aren't supposed to use it :smalltongue: ! They're trying to stop it from being used! The bad guys are trying to assemple it, and their capturing midgard dwarves (the race that built it in the first place) to tell them how to put it back together, and hoe to activate it.I don't assume for a moment that you are a junior DM. But that bolded part right there? Never give the players access to something you don't want them using, or keeping. They are going to want that. Guard it well.

[ahenobarbi]

I'm wondering how you fit six squares exactly into a larger square? Bending space? And six diagonals?

And at least from my reading of the OP, that math missed a step or two, and the actual number would be much larger.

As I read it:

We have a cube.
Each of the six sides of that cube has six puzzles.
Each puzzle is a 7x7 grid, with on blank square, that form different runes.
These are divided into six rows, and six columns, each with eight(?) degrees of freedom, that can be set to four positions. I'd simplify that to agree with Chronos and say it basically allows you to set the tiles to whichever location you want.

Assuming the contents of the 7x7 boxes can be mixed at will, that gives the 6912 tiles 7056 different possible positions each. It's too late here for combinatorics, but as a simple permutation of 6948 items (adding in the 36 blank spaces), gives 1.213e23678 combinations. A number over twenty-three thousand digits long (over 230 googol!). So yeah, it'd be pretty hard to solve without divination magic.

Of course, the above answer is probably off too, but I think it might be a little closer than the one above. I think you'll have to clarify a little better how the cube is made up.


Hmm I'd calculate it this way:

Choose 35 pieces for each side (leaving one empty), for each of 6 sides of the cube =( 35*6)!/35! * (35*5)!/35! * (35*4)!/35! * (35*3)!/35! * (35*2)!/35! * 35!/35!
For each side you have 36! possible arrangements of tiles, multiply result from previous step by (36!)^6
Each tile can be in one of 4 orientations, multiply previous step by 4^(35*6)
Ask Wolfram alpha for help
Enjoy knowledge that the answer seems to be

37310673678616000744604184489069760723822799709331 35425842094096951592540596469383775566022887294747 02001145004163992921931875295244609121487913326911 95156350267759946268156735361341641093490025360380 88982491956045740670937174993650231732358508773935 37235973170734384565355848466843946569124206160751 51653627625637044062223110205734719983150923965799 41689660370906330164792633581795621244455621556667 18246690032491273086194788522465001099435926206016 58853260412420897889686265624825990431324388304945 10412727203897295232729954449137320157647894201920 33123950312425034090709748174342329842564588402127 84185615506773955202467778698470890859256386062470 47478432676111171248368085136756539152980280531430 89407545718192918282926205401174029039401609926521 63277470201463778954547208214003718366649481158157 51198429364533808254573400304609512894412430612752 55115209019255135117210106470648429798832870645868 86540245014892579174705341195861369047898390838440 25996124561697099947446958542576419185368363230261 65209050738921902856659562326353572876461705551367 13410365617457742958711498522574785157668939886224 84033141682454204758599040128591714039011634174912 66708720715019322301621279742996481322232468609288 58415192875506534024231392703202375691066177680440 29542400000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000

or

3.731067367861600074460418448906976072382279970933 135425... × 10^1439

if you don't mid sacrificing accuracy for clarity

[Chronos]

You don't have to leave one blank on each side: The blanks can be moved from side to side via the Rubik moves.

And on thinking about it some more, you can get six diagonal slices on a cube, if you mean edge-to-edge slices (so a slice cuts the cube into two triangular prisms). I was thinking slices between the vertices, of which there are 4.

[ahenobarbi]

You don't have to leave one blank on each side: The blanks can be moved from side to side via the Rubik moves.

I thought this


...each of the tiles can be broken up along all these different lines, and "snap" into place once the turn is completed so every side always has the same number of complete tiles.

meant each side must have the same number of tiles (and the same number of empty spaces) at all times.

[Absol197]

I thought this



meant each side must have the same number of tiles (and the same number of empty spaces) at all times.

You are correct: the mechanisms inside the cube won't let it turn unless all the blank space are in the same spot on all affected sides.

[Madcrafter]

You are correct: the mechanisms inside the cube won't let it turn unless all the blank space are in the same spot on all affected sides.

That complicates things even more, but wouldn't actually affect the number of possibilities.

And nice idea on the diagonals Chronos, but you'd have to make an arbitrary decision on how the tiles that would be "cut in half" would move. Overall, it would probably just make the whole system easier to solve.

[Slipperychicken]

If the PCs aren't supposed to solve it, why bother giving it rules? Just say it's a puzzlebox whose solution is beyond mortal ken. Granted, that's total BS, but it's a start. Maybe it requires some divine/special ability to operate it properly.

[Snowbluff]

I had guessed Rubik posted this thread to write about a cube he made with lots of gold. I was wrong. :smalltongue:

[unseenmage]

I had guessed Rubik posted this thread to write about a cube he made with lots of gold. I was wrong. :smalltongue:

Hey, at least I wasn't the only one.

[Snowbluff]

Hey, at least I wasn't the only one.

Is it bad that I read Rubik's name more than I use it in any other context? I am about as likely to solve one as he is trying to ascertain the meaning and purpose of my name. :smalltongue:

[unseenmage]

Is it bad that I read Rubik's name more than I use it in any other context? I am about as likely to solve one as he is trying to ascertain the meaning and purpose of my name. :smalltongue:

Same here. Though I've solved one. Once, when I was a kid. Strangely now I'm in my thirties and probably couldn't solve one for the life of me.

And also, not sure I want to know the meaning and purpose of your name...

For the thread seconding this,


If the PCs aren't supposed to solve it, why bother giving it rules? Just say it's a puzzlebox whose solution is beyond mortal ken. Granted, that's total BS, but it's a start. Maybe it requires some divine/special ability to operate it properly.

[Snowbluff]

Same here. Though I've solved one. Once, when I was a kid. Strangely now I'm in my thirties and probably couldn't solve one for the life of me.

And also, not sure I want to know the meaning and purpose of your name... Snowblind is a (n AWESOME) name of a song, and I will always do my best to avoid legal issues. Snowbluff is a term for something akin to a dune but using snow. They are a pain to traverse.


For the thread seconding this,
Thirding that.

[Aurenthal]

Is the cube suposed to look like this? (Assume that it has runes and artifacty symbols on the sides)
THIS!!! (http://www.amazon.com/V-Cube-7-Multicolor/dp/B001PGWDSU) Exclamation marks added for emotion

[Slipperychicken]

Is the cube suposed to look like this? (Assume that it has runes and artifacty symbols on the sides)
THIS!!! (http://www.amazon.com/V-Cube-7-Multicolor/dp/B001PGWDSU) Exclamation marks added for emotion

BRO DO YOU EVEN RUBIX
http://www.superliminal.com/cube/cube.gif