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 Post subject: Puzzle Difficulty
PostPosted: Tue Aug 17, 2004 9:21 pm 
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What do you think of adding a difficulty rating to the puzzle descriptions? Maybe having a scale of 1-9 with a 3x3x3 being a 5. That way people would have a little reference when it's time to buy a new puzzle. I know peoples opinions of difficulty for a certain puzzle may vary slightly but it would be nice to have a reference point to give people an idea of what to expect.

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 Post subject: Re: Puzzle Difficulty
PostPosted: Wed Aug 18, 2004 9:23 am 
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skeneegee wrote:
What do you think of adding a difficulty rating to the puzzle descriptions? Maybe having a scale of 1-9 with a 3x3x3 being a 5. That way people would have a little reference when it's time to buy a new puzzle. I know peoples opinions of difficulty for a certain puzzle may vary slightly but it would be nice to have a reference point to give people an idea of what to expect.


Adding in a field is a 10 minute job, but populating 1000+ puzzle entries is the hard part. How do you suggest we come up with the value?! As you've pointed out, it's wide open for personal experience. Adding in a system that averages all data might work, but only if people systematically go through hundreds upon hundreds of puzzles placing their votes.

Maybe an article which rates each basic type of puzzle (probably based on data collected from a series of forum polls) would be more managable.

Does anyone else have any thoughts on any of this?

Sandy


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PostPosted: Wed Aug 18, 2004 1:20 pm 
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what do you think of these preliminary ratings? Just most of the puzzles have.

scale 1-9, 9 being hardest

Rubik's cube.............5
2x2x2....................4
Chex cube(supergroup)....6
Rubik's Calander.........1
4x4x4....................7
Pyraminx.................4
Pyramorphix..............5
Rainbow cube.............2
Homer (heads)............4
Orb(-it).................3
Rubik's Shells...........1,3,7
3 Level Puzzle...........3
Missing Link.............3
Whip-it..................2
Rubik's UFO..............6
Rubik's Dice.............3
Duomaster Masterball.....2
Geomaster Masterball.....7
K-8 Ball.................4
Square 1.................8
Rubik's World............7
Alexander's Star.........6
Megaminx.................7

That's just my opinions, I would love to hear why someone disagrees.

-mike

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PostPosted: Wed Aug 18, 2004 3:12 pm 
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Looks good,

You have several 1's but no 9's. What is the most difficult puzzle? Anyone care to share what they think is the most difficult?

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PostPosted: Wed Aug 18, 2004 3:35 pm 
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I was thinking crossteaser or great gears might be a 9 because I haven't been able to solve them yet, however once I figure them out my opinion may change.

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PostPosted: Wed Aug 18, 2004 3:38 pm 
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skeneegee wrote:
I was thinking crossteaser or great gears might be a 9 because I haven't been able to solve them yet, however once I figure them out my opinion may change.


That pretty much makes my case about personal preference!

There are a few old threads around here somewhere in which people rated various puzzles. I wonder if it's worth digging those up? Perhaps tomorrow when I have a whole new day ahead of me.

Sandy


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PostPosted: Wed Aug 18, 2004 4:05 pm 
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Again, with the personal preferences.

I think the skewb is hard as hell, because I've only toyed with mine for about 5 minutes. I lost interest and it is still unsolved.

Also, I personally think the 5x5x5 is easier than the 4x4x4, since it has fixed centers. The parity problems are much harder to catch on the 4x4x4 also, depending on your method.

...then again, you only have 4 "center pieces" per face to worry about on the 4x4x4, as opposed to 9 on the 5x5x5. Intuitively it's easier to solve the 4x4x4 centers...

...then again, the 4x4x4 doesn't really have centers :roll: you have to make them for yourself. and if you don't know your cube's color scheme by heart (although most people who will be rating the 4x4x4 probably will) and can't figure out to just use the corners to figure out the color scheme, the centers can be a problem.

Just my $0.02.

Doug


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 Post subject:
PostPosted: Thu Aug 19, 2004 12:47 am 
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Dodd wrote:
You have several 1's but no 9's. What is the most difficult puzzle? Anyone care to share what they think is the most difficult?


I missed Mefferts Bandaged Cube. I eventually solved that thing, but it took me three month. The Fisher-2-Cube belongs in the same category. Has anybody else experiences with that puzzle?


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 Post subject:
PostPosted: Thu Aug 19, 2004 1:32 pm 
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Overall, I think this will be great if we get it going. One other thing just came to mind. Some collectors will obviously have more experience solving twisty puzzles than newbie puzzlers. I think we should group all of the supercubes into one category, so votes influence them all. Same with all 3x3x3 non-super/picture cubes, 4x4x4's, 5x5x5's, etc.

My reasoning: someone with a smooth-tiled Rubik's Deluxe is IMO probably going to rate a 3x3x3 as being easier than someone who has just gone and bought a "Magic Cube" at the dollar store, mostly because it seems like people who own a smooth-tiled deluxe are more likely to be experienced speedcubists (correct me if I'm wrong) or puzzle solvers than someone who is rating a Super Cube or Magic Puzzle or [insert generic cube puzzle name here]. Not that it really makes a difference, but I think a more standardized rating scheme per puzzle type (3x3x3, 3x3x3 super, etc) would be better overall :wink:


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 Post subject:
PostPosted: Fri Aug 20, 2004 9:55 am 
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http://chrono.org/other/rcfaq009.txt

Go to section 4.1 which asks the question "how hard is it to solve a puzzle?" and you'll get a couple of "difficulty" lists.

As you all know the answer varies based on the talents and experience of the individual solver. Some puzzles can be easily solved almost by random twisting (Rainbow Cube) while others can be solved intuitively (Masterball) and still others require some group theory math to figure out (5x5x5).

I rate a standard 3x3x3 cube much higher than a 5 because with the exception of a few very attuned people, one cannot just solve it without sitting down and working out mathematical solutions.

And using existing algorithms to swap pieces doesn't count. :wink:

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 Post subject:
PostPosted: Fri Aug 20, 2004 11:46 am 
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Hmmm... interesting...

I would think that one would need to take several things into consideration when ranking the difficulty of puzzles and try to 'discourage' the personal pref factor..

1. Possible perumtations (more perms gives higher score)
2. Axis of rotation (more axis gives lower score)
3. Sticker supergroups (cubie alignment = higher score)
4. Morphs (morphers would get a higher score)
5. Parity issues (no parity issues = lower score i.e dino cube)

the scores would be added together and your scale should be set... of course there could be other criteria too...

just a thought...
_pink


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 Post subject:
PostPosted: Fri Aug 20, 2004 12:56 pm 
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I was just thinking that the scale could be used by people who are planning to buy a new puzzle for reference as to the difficulty they should expect. Although I agree the Rubik's cube probably deserves to be higher than 5 on any other scale, I thought this scale could be based on the Rubik's cube. Since anybody who is a member here has played with Rubik's cube, it would be a good basis to see if a certain puzzle is more or less difficult than Rubik's cube and by about how much.

For instance "NoobCollecter84" wants to buy a new puzzle and is disappointed with his/her last purchase because it was far easier/harder than expected. S/he simply references the puzzle database and together with the Price Tracking System (which is an awsome feature btw) decides to buy something they were looking for.

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 Post subject:
PostPosted: Fri Aug 20, 2004 1:11 pm 
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Hmmm, I see where you're going Skeneegee. In conjunction with this 'scale' I think there should be an availability list... we all know how difficult some of these little gems are to come up with...

_pink


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PostPosted: Fri Aug 20, 2004 1:46 pm 
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> I think there should be an availability list...

Something very similar to this is "in the works" already. Using the Collection Manager data, I'll be setting up a system which calculates how common each item is. Naturally, because of the specific data used, the final statistic will be a little skewed. However, I think the conculsions you will be able to draw from examining the data will be just as useful (if not moreso) than an arbitrary ranking system based merely upon personal experience.

Sandy


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PostPosted: Sun Aug 22, 2004 12:19 am 
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maybe it should be rated on how easy it is to learn how to solve?

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 Post subject:
PostPosted: Mon Oct 25, 2004 2:00 am 
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Indead a personal opinion

I would rate from my experience:
Pyraminx.................3
Rainbow cube.............1
Rubik's UFO..............4
Geomaster Masterball.....4
K-8 Ball.................4
Square 1.................7
Rubik's World............6
Alexander's Star.........3
Megaminx.................5
Missing Link.............1

Skewb 4
5x5x5 7
have no idea why the megaminx would be harder as the Rubik's cube, I use about the same basic method to solve it as the Rubik's cube.As for the the Rubik's UFO and the masterball, they are the same type of puzzle, so they should rank the same.

The Rainbow cube is evently a 1

I would rate the Fishers 4x4x5 as a 8

If you can solve a 3x3x3 and a 4x4x4 you can solve a 5x5x5 so thet should be the same

A 9 puzzle ....


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 Post subject: Hard to determine relative difficulty
PostPosted: Wed Apr 06, 2005 8:05 pm 
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It is very hard to determine relative difficulty to similar puzzles.

It is hard for me to judge the 5x5x5 because I was trying to master the 4x4x4 when I bought it.

Maybe the main puzzles could each be linked to a difficulty forum ;)


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 Post subject:
PostPosted: Mon Nov 21, 2005 9:55 am 
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Hi to everyone...!

Just to state a possible way of "counting" the difficulty of each puzzle.


Note that each puzzle can be distinguished mainly by two things:

(1) The total combinations (without the trivial ones),
(2) Its generators.

For example, Rubik's cube has a known number of 4.3*10^19 total (non-trivial) combinations, and (by stabilising one element) we obtain six generators of order four.

The "difficulty number" D of the puzzle may then be determined by the formula:

D = C/R

where
C is the number of total combinations
R is the product of all the orders of the generators.


The reasoning for all the above is that by having more generators (e.g in the Rubik's Shells case), it is easier to solve the puzzle.
Moreover, a higher order per generator (with the total number of combinations already known) gives more ways of solving a puzzle.

Such generators need not necessarily be independent, but the presence of more of them of a higher degree, gives more options in finding a solution.

Some puzzles may seem exotic (e.g the Astrolabacus), but it's huge generators make things much easier than expected...! :wink:
(just imagine an Astrolabacus - which please note, is such a wonderful masterpiece - with same number of marbles, but smaller tunnels... wouldn't THAT be a challenge?)

Below are some experimental numbers showing the "difficulty" parameter D in millions:

Rubik's 2x2x2: 0.23
Rubik's 3x3x3: 10,498,046,875
Rubik's 4x4x4: 2.8*10^34
Rubik's Shells (no buttons pressed): 115,361
Rubik's Shells (both buttons pressed): 7,383,099
Astrolabacus: 1057.641

Now if you want to use some other way or metric, then why not? 8-)
I am not sure if solving the 2x2x2 cube is easier than solving the Rubik's Shells (no buttons pressed), but I hope you like this preliminary suggestion...!


Peter

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 Post subject:
PostPosted: Mon Nov 21, 2005 10:53 am 
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Interesting, Peter. But maybe a logarithmic scale would help, too, as those numbers range over many orders of magnitudes.

While we're tossing metrics around, how about

T = log C / log N,

where C is the same as yours, and N is sum of (order-1), taken over all generators?

Without wanting to think too hard about it right now, I guess that T would be (very roughly) the solution length, assuming that not too many generators commute, and that the number of generators is not too small (as in the Shells with both buttons pressed). But solution length is probably a poor measure of difficulty; for example, the Megaminx is no harder than the 3x3x3 but takes more turns to solve -- the extra length of the Megaminx solution is just "more of the same" (ad nauseam), not extra difficulty.

Unfortunately, the difficulty of puzzles is highly subjective, and can depend on mathematically irrelevant factors. For example, many people might find the Braintwist significantly harder than the (equivalent) Dino Cube, simply because you have to flip the puzzle to see both sides of the pieces, and to make the second set of corner twists possible. Flipping that puzzle makes it easy to lose one's bearings...

Similarly, the Ultimate Skewb might be tricky when you first meet it, but mainly because of its unusual shape, and the presence of left- and right-handed pieces with the same colours on them.


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 Post subject:
PostPosted: Mon Nov 21, 2005 12:01 pm 
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Thanks for the reply Mike. You are raising a lot of interesting issues! :)


Mike G wrote:
Interesting, Peter. But maybe a logarithmic scale would help, too, as those numbers range over many orders of magnitudes.

While we're tossing metrics around, how about

T = log C / log N,

where C is the same as yours, and N is sum of (order-1), taken over all generators?



Indeed, it sounds more realistic and log-ical! :P




Mike G wrote:
Without wanting to think too hard about it right now, I guess that T would be (very roughly) the solution length, assuming that not too many generators commute, and that the number of generators is not too small (as in the Shells with both buttons pressed). But solution length is probably a poor measure of difficulty; for example, the Megaminx is no harder than the 3x3x3 but takes more turns to solve -- the extra length of the Megaminx solution is just "more of the same" (ad nauseam), not extra difficulty.


Exactly! I believe that finding a way to describe a Megaminx having a smaller D than a 3x3x3 cube, could be a very good starting point.
A megaminx, just like a Dogic are not as hard, but they have one thing in common: many combinations and many generators!!!





Mike G wrote:
Unfortunately, the difficulty of puzzles is highly subjective, and can depend on mathematically irrelevant factors. For example, many people might find the Braintwist significantly harder than the (equivalent) Dino Cube, simply because you have to flip the puzzle to see both sides of the pieces, and to make the second set of corner twists possible. Flipping that puzzle makes it easy to lose one's bearings...


Very true. But it is also true that getting familiar with an exotic looking puzzle is independent with respect to the maths behind it.
I know what you mean, and I kind off ommitted the fact, because it's like learning bicycle while cycling inside a maze...!
You can learn how to always cycle, but the maze is still there to confuse us LOL



Mike G wrote:
Similarly, the Ultimate Skewb might be tricky when you first meet it, but mainly because of its unusual shape, and the presence of left- and right-handed pieces with the same colours on them.


When I first tried to solve a Square-1 cube I was badly stuck for a while. So, how about adding some "asymmetric" factor (maybe depending on the different shapes) for those cases? :wink:


Thanks for the interest, it will be great if such a formula can be devised! :D




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 Post subject:
PostPosted: Mon Nov 21, 2005 1:54 pm 
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kastellorizo wrote:
I believe that finding a way to describe a Megaminx having a smaller D than a 3x3x3 cube, could be a very good starting point. A megaminx, just like a Dogic are not as hard, but they have one thing in common: many combinations and many generators!!!

Megaminx and Dogic have other things in common: the generators affect relatively few pieces (they are "shallow cut" puzzles), and the number of pieces shared by a pair of adjacent faces is relatively small. This double dose of "shallowness" makes it particularly easy to construct useful commutators... contrast with the Pentultimate, which, despite having six generators, would probably be hard to solve.
Quote:
So, how about adding some "asymmetric" factor (maybe depending on the different shapes) for those cases? :wink:

Hmm... maybe you have an idea of how to do it already? I don't. :wink:

If we ignore the parity problem and the shape restoration for Square-1 (though those factors are rather important), the changing shapes don't add much to the difficulty of Square-1. Any remaining difficulty comes from the fact that U, D and / (flip) affect a large proportion of the pieces.

So... just as a starting point, it would be nice to have a rough measure of "shallowness".

This problem of quantifying difficulty used to worry me, but I eventually gave up thinking about it: we can think of factors that help to make a puzzle difficult, but we don't know for sure how to combine the factors, because we are trying to predict a human response to the puzzle. One possible way around the problem would be to design a formula incorporating these factors, but with a small number of adjustable parameters in it. You'd choose the parameters in the formula to obtain a reasonable fit to the ordering by difficulty, as determined by the results of a large poll. That way some of the subjective, human element could be allowed to contribute.


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PostPosted: Mon Nov 21, 2005 11:02 pm 
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Mike G wrote:
Megaminx and Dogic have other things in common: the generators affect relatively few pieces (they are "shallow cut" puzzles), and the number of pieces shared by a pair of adjacent faces is relatively small. This double dose of "shallowness" makes it particularly easy to construct useful commutators... contrast with the Pentultimate, which, despite having six generators, would probably be hard to solve.


Shallowness does help to make things easier, and this is the case where the generators are independent from each other.
Therefore, one of the next things to do, is to take into account the amount of "common element changing" shared by generators (acting on the elements as a permutation).
And after this is cleared, then and only then, we may supply a better formula!



Mike G wrote:
Quote:
So, how about adding some "asymmetric" factor (maybe depending on the different shapes) for those cases? :wink:

Hmm... maybe you have an idea of how to do it already? I don't. :wink:


Would multiplying the D number with the different types of shapes (where permutations are feasible) help?
LOL I know I don't give up easily...! :P



Mike G wrote:
If we ignore the parity problem and the shape restoration for Square-1 (though those factors are rather important), the changing shapes don't add much to the difficulty of Square-1. Any remaining difficulty comes from the fact that U, D and / (flip) affect a large proportion of the pieces.

So... just as a starting point, it would be nice to have a rough measure of "shallowness".



yeap, agreed!




Mike G wrote:
This problem of quantifying difficulty used to worry me, but I eventually gave up thinking about it:


No! Never give up! LOL
You have clearly got some experience on this matter plus some technical knowledge... we need you!!!



Mike G wrote:
we can think of factors that help to make a puzzle difficult, but we don't know for sure how to combine the factors, because we are trying to predict a human response to the puzzle. One possible way around the problem would be to design a formula incorporating these factors, but with a small number of adjustable parameters in it. You'd choose the parameters in the formula to obtain a reasonable fit to the ordering by difficulty, as determined by the results of a large poll. That way some of the subjective, human element could be allowed to contribute.


It is true that a poll can help, different people see things differently, but in a mathematical and geometrical point of view, I believe we can come up with something!
Maybe something not perfect and 100% accurate, but which can possibly determine some averaged way of giving an idea to a normal person of what to expect...!
(I hope I am making sense here LOL)

Then after getting such a formula out, we could use the experiences of people here, to verify the "success percentage" of the formula! :)

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 Post subject: rating difficulty
PostPosted: Wed Nov 23, 2005 8:44 pm 
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Hi Peter, Mike(s), Sandy et al,

The reasoning seems headed this way:
If there are 43,252,003,274,489,856,000 possibilities, and if it takes, say, 20 turns to reach any of the possible positions.
Then (18 * (15^19))/43,252,003,274,489,856,000
= 9225.71851937566816475041943268785

Similarly for the the 2x2x2 - 3,674,160 positions reached in 11 moves would be

(9 * (6^10)/544195584
= 148.114285714285714285714285714286

3x3x3 ~ 9225
2x2x2 ~ 148

Then maybe scale these numbers:
cube root of 9225 is nearly 21
square root of 148 is a little over 12.

If the difficulty of the 3x3x3 is about 21 then the difficulty of the 2x2x2 is about 12. That seems fairly reasonable to me, speaking as someone who solves them himself with no outside help.

Does this seem reasonable to you?

Anyone care to continue this for the 4x and 5x? I'm tired and don't know the diameter of these puzzles.

David J


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 Post subject: Re: rating difficulty
PostPosted: Thu Nov 24, 2005 6:58 am 
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Hi David J

David J wrote:
If the difficulty of the 3x3x3 is about 21 then the difficulty of the 2x2x2 is about 12. That seems fairly reasonable to me, speaking as someone who solves them himself with no outside help.

Does this seem reasonable to you?

The numbers seem entirely reaonable, but I'm just not sure about the motivation behind them: to calculate them you are essentially using the "redundancy" of sequences of length 20 and 11, in excess of the number of states.

Of course, the diameters (or estimated diameters) of the puzzle groups are numbers somewhat similar to yours in this case... but I feel that there should be some well-defined, intrinsic measure of group "complexity" (not simply the diameter), given a set of generators for the group.

I also feel that I'm not well-equipped to think seriously about this problem. ;)

Mike


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PostPosted: Thu Nov 24, 2005 10:21 am 
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David, I see your point for taking into account the number of moves to reach a certain position. The only thing here is that we may be knocking on the doors of some unsloved problems, which may make the use of a formula not very practical...
Still, what I said are not words carved on stone, there still may be a way!

I actually consider your thought as a totally different method for measuring the "difficulty" of puzzles. So far proposed methods:


- (1) Empirical, where experienced players may submit their difficulty grade,
- (2) Combining the possible combinations of a puzzle with its generators, their orders, and their "shallowness" (i.e. common interaction with other generators)
- (3) Using the number of moves where certain states are reached.


Hmmm... I don't know, but in my humble opinion, it seems we got a cute project here...! :P


Peter

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PostPosted: Thu Nov 24, 2005 11:22 am 
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How do you guys take into account people who for example think the 3x3 is easier than the 2x2?


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Would a comparative difficulty system work?

A poll system could be made to compare the difficulty of puzzles relative to each other.

Instead of giving a puzzle an actual difficulty, the result could be "60% find the 2x2 easier than the 3x3", etc.

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It is true that no system is 100% perfect. The proposed system with the generators will attempt (always in a cruel mathematical numbering way) to compare combinations and possibilities.

The "shallowness" (i.e. the degree of interaction between generators) is a safe way to go, in the sense that the more independent the generators are, the easier the solution is. This is obvious. The difficulty is to determine the formula that expresses it (which may not be *that* difficult after all).

Human personal factors, like the reasons someone finds a puzzle is difficult, or they way he is psychologically confused by certain shapes, are very hard to include, at this moment at least.
(maybe my sister can assist after she completes her psychology degree LOL)

Like said, it is not going to be a 100% accurate way to determine difficulty, but it will certainly should give a "good idea" of how difficult its maths is.

Just look at it as something similar to the FIFA football ranking, but regarding puzzles! :P

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PostPosted: Thu Nov 24, 2005 2:42 pm 
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kastellorizo wrote:
Just look at it as something similar to the FIFA football ranking, but regarding puzzles! :P

Sure, that aspect is not too hard, even if each person polled is asked only for the ordering by difficulty of the subset of puzzles they actually have an opinon on.

Each person's ranking of the puzzles can be thought of as a single tournament between some of the puzzles, in which each puzzle wins, loses (or perhaps draws) with each of the others. To get a final ranking you might use something like the Glicko system, which (unlike Elo) attempts to give a "reliability" statistic, in addition to a "rating level". You would expect the "reliability" to be worse for the rare (or very difficult?) puzzles, as not many people would have an opinion on them.


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PostPosted: Thu Nov 24, 2005 3:08 pm 
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kastellorizo wrote:
The "shallowness" (i.e. the degree of interaction between generators) is a safe way to go, in the sense that the more independent the generators are, the easier the solution is.

Might have to be careful about what we mean by this interaction. For example, the generators for the Clocks don't "interact" at all, if you work with the right variables for that puzzle. :D

Of course, there is overlap between the sets of clocks affected by each generator, so "the right variables" are clearly not the sensible ones to use here. The individual clock faces are more obvious and natural...

Oh, darn. I'm getting too interested in this now. :(


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PostPosted: Thu Nov 24, 2005 10:40 pm 
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Mike, I agree with the "reliability test" thought of ours. I am sure this topic has been discussed before, I hope I can give a new way of thinking of it! :)
After some of us start showing some formulas, time and only time may tell
which one stands best compared to the other formulas. And if it behaves "well" for the rare ones, that would be even better!



Mike G wrote:
kastellorizo wrote:
The "shallowness" (i.e. the degree of interaction between generators) is a safe way to go, in the sense that the more independent the generators are, the easier the solution is.

Might have to be careful about what we mean by this interaction. For example, the generators for the Clocks don't "interact" at all, if you work with the right variables for that puzzle. :D

Of course, there is overlap between the sets of clocks affected by each generator, so "the right variables" are clearly not the sensible ones to use here. The individual clock faces are more obvious and natural...


Exactly! This is the key point, where each other's formulas are going to differ! Now I got to do some thinking! :wink:
Of course those who have some sort of science degree may have an unfair advantage...!



Mike G wrote:
Oh, darn. I'm getting too interested in this now. :(



LOL I hope more get interested in this too! :)

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 Post subject: Re: rating difficulty
PostPosted: Fri Nov 25, 2005 4:36 pm 
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Hi Mike G.,

Mike G wrote:
Hi David J

David J wrote:
If the difficulty of the 3x3x3 is about 21 then the difficulty of the 2x2x2 is about 12. That seems fairly reasonable to me, speaking as someone who solves them himself with no outside help.

Does this seem reasonable to you?

The numbers seem entirely reaonable, but I'm just not sure about the motivation behind them: to calculate them you are essentially using the "redundancy" of sequences of length 20 and 11, in excess of the number of states.


The motivation? Do you mean the logic?

If there are multiple ways of doing the same thing then finding any one of them becomes easier.

Mike G wrote:
Of course, the diameters (or estimated diameters) of the puzzle groups are numbers somewhat similar to yours in this case...


That's what I was after.

Mike G wrote:
but I feel that there should be some well-defined, intrinsic measure of group "complexity" (not simply the diameter), given a set of generators for the group.


As an example of the complexity as I see it, the last level can be solved permuting corners, orienting corners, permuting edges, and orienting edges. These four actions may be done in any sequence or in combinations like OLL-PLL, and CLL-ELL. This redundancy *is* in the math I posted above and *is* intrinsic to the cube.

Mike G wrote:
I also feel that I'm not well-equipped to think seriously about this problem. ;)

Mike


I don't know what you mean. Is this a reflection of how many algs you have found on your own?

Cheers,
David J


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PostPosted: Sat Nov 26, 2005 12:55 am 
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David, are there any references for the formula (m-2)*(m-5)^(m-1)/[combinations] that you are using?
(did I even display it correctly? LOL)

Or is it and empirical result? :)

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 Post subject: Equation
PostPosted: Sat Nov 26, 2005 6:20 pm 
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Hi Peter,

kastellorizo wrote:
David, are there any references for the formula (m-2)*(m-5)^(m-1)/[combinations] that you are using?
(did I even display it correctly? LOL)


Where you have (m-2) I have 18 moves which are possible. Each of the six sides, F L U B R D, may be turned three ways - 90 degrees clockwise, 90 degrees counterclockwise, or 180 degrees. Six sides times three turns and all that jazz.

Where you have (m-5) I have moves which do not involve the side just turned. Five sides times three turns...

I think I read of that approach in the early 80s in a cube newsletter.

Some might say that it should be 18^20 for the 3x3x3.
12748236216396078174437376/43252003274489856000
=294743.254676368274582560296846011
Cube root = 66.5499844431978318535663020882689

So then the 2x2x2 would be 9 ^11 = 31381059609
/ 3674160
= 8541.01607142857142857142857142857
square root = 92.4176177545632877506767640974661

Which might please Stefan. 8-)

kastellorizo wrote:
Or is it and empirical result? :)


Whoa, Dude! I've been busy, but not *that* busy! LOL

David J :)


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 Post subject: Re: Equation
PostPosted: Sat Nov 26, 2005 11:17 pm 
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David J wrote:
Which might please Stefan. 8-)


Yeah :D. Though, the two main reasons I see why 3x3 can be easier are:

- You have the centers as reference. Newbies confuse themselves all the time because they rotate the cube as a whole and then they don't know where they were and start all over again. At least the 3x3 gives them the chance to hold onto something that never changes.

- It's much easier to achieve something impressive with the 3x3. One face is fairly easy, one layer is still easy and even F2L is doable for newbies if they really try. That's 12 pieces not even counting the centers! Ok, the last layer is of course hard because you don't have much freedom anymore, but at least they can get something done which can spark the motivation to go even further. And I claim the most important reason why people can't solve cubes is missing motivation (and interest/opportunity)...

So yeah, it's psychological. Try to put that into a formula :D

Cheers!
Stefan


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PostPosted: Sun Nov 27, 2005 8:05 am 
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Psychological questoinaire/review... to be completed by ALL Twisty puzzle forum members!
(A penalty equal to a curse of getting finger crambs during competitions applies for those who don't...!)


(1) When visiting relatives, do you always (frantically) search the map for the shortest path?
(2) Do you use plain sugar with spoon, or only sugar cubes?
(3) Which number is larger, 2 or 3, and why? (this question only applies to Stefan LOL)
(4) We may not square the circle, but can we cube the sphere?
(5) Which one appeared first on planet earth, the Astrolabacus or the Australopithecus?



All the above data will remain confidential and used for a secret difficulty formula, unless someone of you becomes too famous, then they can be used for blackmail..! :P

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 Post subject: Re: rating difficulty
PostPosted: Mon Nov 28, 2005 7:02 am 
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Hi David J
David J wrote:
If there are multiple ways of doing the same thing then finding any one of them becomes easier.

OK, in my eyes that's "motivation". Thanks for explaining.
Quote:
Mike G wrote:
I also feel that I'm not well-equipped to think seriously about this problem. ;)
I don't know what you mean. Is this a reflection of how many algs you have found on your own?

No, I simply feel that I don't know enough group theory. If there is a well-defined measure of complexity (other than the diameter), given a group and a set of generators for it, that measure is likely to be known already.

Of course, that needn't stop us having fun.

Mike


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PostPosted: Tue Dec 20, 2005 1:55 am 
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I plan to spend some time on this (but not too much) and maybe have some generalised (and vague) formula, hoping that a 70% of puzzlers will agree.
(then it will be acceptable in my book LOL)

Anyway, right now must finish some last urgent jobs before I leave for holidays..! :wink:


Peter

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 Post subject:
PostPosted: Sun Jul 23, 2006 3:34 am 
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I disagree with most people putting the Square - 1 high up, I found average to solve, the skewb in my opinion is harder.

I don't think this would work very well, different people's minds work in different ways and therefore can solve different puzzles with ease, they maybe someone who can solve an astralabacus (SP?) but cannot do a 2x2, I understand you are just trying to get a brief overview but I think it will be too difficult to rate them.

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PostPosted: Sun Jul 23, 2006 11:15 am 
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should probably rank puzzles by how difficult it is to solve without a method.


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PostPosted: Sun Jul 23, 2006 12:35 pm 
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How do you judge that though?

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PostPosted: Sun Jul 23, 2006 2:02 pm 
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hmmm, trying to find out that article i had read, that was very similar to this. i think it was on Jerry Slocum's page...

it had to do with the difficulty of burr type puzzles.

now, everyone has their own preference as to what's easy and what's not. what has been solved easy to someone else may be hard to another.

what they did decide to do, was to analyze the puzzle itself and break it down.

so for example a normal 6 piece burr -
if it took 5 moves to get the first piece out then 3 more to take the second piece out, and 2 more for the puzzle to fall apart, it would be labled something like 5.3.2

it was quite interesting once they all agreed to order the puzzles like that.

if there were something we could do to disect and analyze twisty puzzles, i'm sure we could get a set order on it :) ---- now it still wouldn't explain how hard the puzzle is, but it would explain how complex it can be. --- then again, i'm sure Japp has done alot of this on his site, explain how many positions the pieces can be, the order of the puzzle, etc.

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 Post subject:
PostPosted: Sun Jul 30, 2006 11:56 pm 
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How about you just make it like a slider-bar poll?

(Sandy, if you want I could program that for you if you dont have time. Im a PHP / MySQL programmer.)

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