The most improbable thing you've ever seen?

[Borodog]

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It's not supposed to look like this?



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I have no idea. As far as I knew you just needed to get each side to have the numerals 1-9. Mine is gray with white numbers, so it may be a different manufacturer.

Edit: Yeah, it looks like there must be more than 1 solved state. If you look at the location and orientation of each numeral, they are all the same. 1 upper right, 2 upper middle, 3 upper right, 4 left middle, 5 in the center, etc.

So now I want to know, how many "solved" states are there; not the "metasolved" state with all the numbers in order, but just solved according to Soduku rules, with each face containing the numbers 1-9.

If we can ascertain that this was not miraculously unlikely, I'll have a lot easier time getting to sleep.


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We could have a problem here... isn't this game played with numbers 1-9 only? I'm thinking your cube may just be in 1 of its many unsolved states [img]/images/graemlins/smile.gif[/img]

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There seem to be many more state where the numberals do NOT appear exactly once on each face than there are states where they do.

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What I was driving at (sorry for the confusion) was are you sure you know exactly what the solved state is supposed to look like

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Yes; I've already conceded that there is a "metasolved" state where the numerals are all in order, and I haven't hit upon it by random. What I would like to know now is how many "simply solved" states there are (like the one I hit upon). But again, off to bed.

[captain2man]

It actually IS much more likely.

My solution only took into account solved states where all the sides had the same exact arrangement.

Presumably - each side could be solved but in a completely different arrangment.

So - each of the 6 sides could be in any one of 9! arrangements - which would mean 6 * 9! total solved states.

This would be 2,177,280 total solved states of the cube.

So now we're down to

43252003274487678720 to 2177280

or

19,865,154,355,199 to 1

We're still on the ultra-rare side here.

Sorry - tried to help.

I don't see how to reduce it further - but maybe others can.

[LuckOfTheDraw]

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There are 9! combinations of how 1 through 9 could be arranged so presumably the puzzle could be in a solved state for each of those 9! unique arrangements.

9! = 362,880.

Assuming your prior "total" possible states is correct, you are

43252003274489493120 to 362880

or...

119,190,926,131,199 to 1 to have a random arrangement of the cube be in a solved state.

Hmmm....not quite as rare as you originally thought...but - um - yeah.....still rarer than almost anything I could imagine.

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It looks to me that a lot of this flawed. First of all, there aren't 9! arangements per face because of the physical constraints of the rubik's cube (e.g. the 9 could never be in the middle position).

Edit: In your 2nd post you made the same mistake of ignoring the physical limitations of the cube. Each face is not independant of the rest.

[Thythe]

Maybe if you post in the probability forum they'd give a shot at it. I certainly can be of very little help...

[WilyTilt]

No probability in this one, but it's the most improbable thing I've seen-

I was an intern in D.C. in fall, 2004, and volunteered for a campaign in Dallas, TX for a few weeks. One day, as I was canvassing a street there, I met a girl from another unrelated campaign group who just got graduated from my college. I chatted with her for a bit, noticing she had a slight speech impediment that gave her a weird accent. We didn't exchange emails or anything.

Two weeks later, I was back in D.C. in my intern job. One night, in a Starbucks, I heard the same same accent again, I looked around and it turned out it was her again. It was complete coincidence, that we would meet in a random street in Dallas and then two weeks later in a cafe in Washington. Of course I was freaked out by the whole we're meant for each other thing, asked her out to dinner, and tried to hook up with her. Unfortunately, it didn't lead anywhere, but if I run into her again I wonder if I should be proposing to her on the spot, Billy Crystal style...

[ChrisV]

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It looks to me that a lot of this flawed. First of all, there aren't 9! arangements per face because of the physical constraints of the rubik's cube (e.g. the 9 could never be in the middle position).

Edit: In your 2nd post you made the same mistake of ignoring the physical limitations of the cube. Each face is not independant of the rest.

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Yeah, I think this is going to reduce a lot further. Looking at the cube, the 5 being in the center of every face is not the only restriction. There's other stuff like the fact that the 4s are, as far as I can tell, always on edge cubes, while 1s and 7s are always on corner cubes. Yes?

This is going to drastically cut down how many ways the cube can be arranged.

[DeuceSeven]

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It actually IS much more likely.

My solution only took into account solved states where all the sides had the same exact arrangement.

Presumably - each side could be solved but in a completely different arrangment.

So - each of the 6 sides could be in any one of 9! arrangements - which would mean 6 * 9! total solved states.

This would be 2,177,280 total solved states of the cube.

So now we're down to

43252003274487678720 to 2177280

or

19,865,154,355,199 to 1

We're still on the ultra-rare side here.

Sorry - tried to help.

I don't see how to reduce it further - but maybe others can.

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You make my head hurt.

[captain2man]

Yes - absolutely....my result is a "maximum" for a cube without those physical limitations.

I'm assuming OP's figure of possible arrangements takes into account the physical limitations as well.

Yes - the problem is reduced much further than my post before this one indicates.

I'd also like to see the Probability forum guys take a crack at this.

[ChrisV]

Also, note that the odds against Borodog solving it randomly need to be cut down HUGELY because the cube was actually one turn away from being solved. So we need to count not only the number of solutions, but also the number of arrangements one turn away from being a solution - NONE of which are going to be solutions themselves, so we won't be double counting any positions.

[wet work]

When I was in HS I worked for the Rec. Council, so I had keys to the local ES, MS and HS. Me and my friends would sneak into the MS on weekends, usually when we were dosing. One of the things we would do was to take out EVERY ball in the gym and go crazy.
Anyway on this one particular night I'm standing at the free throw line and decide to take a shot at the opposite hoop. BOOM! Perfect swish. All of my friends just freak out. I look over and say "That's nothing, check this out."
Boom! Another perfect swish. Everybody was speechless. Definitely the sort of thing that only seems to happen when you're tripping. There was no way possible I wasn't going to try for three, but I missed of course.