Pyraminx - Unscrambled tips (fairness) ?

[Radu]

Hi. I posted this first on WCA forum, but due to the lack of activity and my curiosity to hear also other opinions I'm posting this here too (slightly corrected):

Sorry if this has been discussed before, but I couldn’t find a topic here or on speedsolving, so please redirect me to one if it exists!

This topic might sound silly, but I think this “problem” is much more interesting than it appears to be at first sight. You just have to think a bit after reading this. All these came to my mind while I was practicing the pyraminx these days and realized I scramble every time all the tips, even if the scramble didn't show that I have to do it. I just want to make it as hard as possible every time I solve it. I asked myself how fair is not scrambling all the tips? And below is the explanation I find most realistic...

What I’m going to explain in this topic is why I think that in a pyraminx scramble all tips should be mixed and not let the actual random state scrambler give us a scramble with unscrambled tips. I’m totally convinced about this, unless we consider a solved puzzle a scrambled one, if the scramble gives us a solved puzzle. Just be a bit open-minded when reading this.

For the beginning, the main point is that the pyraminx can be regarded as 2 puzzles in 1…we just don’t see it like this. It’s the puzzle scrambled by the big letters (U, D, R, B), the main body and the puzzle were we scramble by the small letters (u, d, r, b), the tips.

Let’s suppose that we have 2 Rubik’s cubes, connected by a stick from one’s center face to the other. We would regard this as one puzzle. Obviously, if we would generate a scramble we would want both cubes to be scrambled and not let one of them in a solved state. (I could have said to imagine a siamese cube, but I wanted a more rudimentary approach). We can now think of 5 or 10 Rubik's cubes connected like this. We would need to scramble all of them to be fair. The same is for pyraminx, with the only exception that one of the puzzles (the tips) is just extremely simple and our mind gets “confused” by this. If we look at a tip and the rest of the pyraminx it’s almost a 1x1x2 cube. You have to twist it, to get it solved. It’s a puzzle with only 3 states, but it's a puzzle itself, not just a random layer. The tip is totally independent of the rest of the pyraminx which is the 2nd puzzle.
So, there are 2 different and independent solves. Leaving the tip unscrambled is like giving a solved puzzle to the solver.

Let’s extrapolate and suppose now another example. We have a huge puzzle with 100 tips. You have to solve the puzzle by bringing all 100 tips to the right position in order to solve this. Would it be fair to give to a competitor the puzzle scrambled with all 100 tips and to another one with only 75? This will obviously become just a matter of luck. It’s like giving someone to solve 100 puzzles and to another one 75!
Of course, the pyraminx is more complex and has also the body which has to be solved by some algorithms and the tips by some others, but technically it is a puzzle with n tips, where n=4. As long as we can scramble a puzzle, why not do it? And why accept unscrambled tips?

So, what I said in the beginning is false. The Pyraminx is not 2 puzzles in 1, but 5 puzzles in 1. We have the main body + 4 tips, which compose the puzzle. Therefore, it would be fair to always scramble all the tips of a pyraminx, if we want to have a fair scramble, otherwise some will solve a different number of puzzles than others. Pyraminx is confusing because it's the only WCA puzzle which is actually made of more puzzles (we just don't see it like this...we regard the tip more like a layer than like a puzzle) and also because n=4 it's a small number and doesn't seem to influence the solve too much. Solving 1 tip is less then 0.5s but it actually could be a puzzle with much more tips like in the example above where it's obvious that all the tips should be scrambled in order to be fair with all competitors.

I don’t pretend the regulations should be changed as we have to be consistent with the scrambles in the past and all the competitors must compete under the same rules, but I just wanted to point this out, see some other opinions about this issue and start a discussion about it. I hope I will not be the only one who sees the problem like this.

Thanks for reading!

 

[Kirjava]

This is the same as the scramble filtering thread.

Only with poorer rationale.

 

[Radu]

This really has nothing to do with that thread. It's strictly about the pyraminx and how we see it as a puzzle. It's about some basic concepts of it. What I was trying to say above about the scrambling concept of the pyraminx and the filtering of easy scrambles in that other topic it's not the point of this thread. It's just an exaggerated comparison.

 

[RNewms27]

If it is more satisfying to call an unscrambled tip a 1-tip skip or 2-tip skip then please do so. If we start scrambling all tips, every previous scramble < three tips would be unfair?

 

[Zarxrax]

I see what you are saying here, but I believe a more interesting approach would be to not scramble tips at all, rather than to ensure that they are scrambled. Solving the tips does not cause any sort of mental effort, its merely a dexterity challenge.
Also, by essentially making the scrambles "harder", it would provide an unfair advantage to past competitors who did not have this restriction.

 

[qqwref]

I think it's definitely a scramble-filtering issue. Scrambles with, say, 2 tips solved are obviously and objectively easier (as a group) than scrambles with 0 tips solved, as you pointed out with your "5 puzzles in one" idea. The questions we each have to answer are:
- is this a problem, in an already luck-based event? and
- if so, should we remove easier (1+ tip solved) scrambles from competitions?

 

[cmhardw]

My first thought is to see how many of the possible scrambled states this restriction would cover.

Including the rotations of the outer tips, the pyraminx has the following number of legal scrambled states:
\( 2^5*3^8*\frac{6!}{2} = 2^4*3^8*(6!) = 75 582720 \)

If you require that all 4 scrambled tips be unsolved at the start of the scramble, then there are the following number of possible scrambled states:
\( (2^5)*(2^4)*3^4*\frac{6!}{2} = 2^8*3^4*(6!) = 14929920 \)

The (Number of legal states with unsolved tips) is approximately 19.75% of the (Number of legal states to the pyraminx - including the tips).

I think that if we make an event using the pyraminx puzzle such that at the start of every scramble all 4 tips are unsolved, then this is really only allowing for about 19% of the total possible states to a pyraminx (including the tips). I think such an event would be interesting, and in a way more "fair" by probably several different definitions of what "fair" could mean in this context. However, I feel that such an event should not be called a "pyraminx" event. Perhaps a "Modified Pyraminx" event, or something to that effect would be a good name. I just feel that only allowing as scrambles approximately 19% of the total possible configurations of the puzzle as a physical object (not an abstract construct) would change the nature of the event so much so that the name of the event should probably also be changed.

That's my take on it.

--edit--
I'm aware that this line of reasoning also means that the name of the "Magic" and "Master Magic" events should probably likewise be changed to "Modified Magic" and "Modified Master magic". I don't think that the names of the Magics events should be changed, so perhaps my line of reasoning is not a very good one for considering the pyraminx, and whether or not all tips should be always scrambled or not.

 

[Tim Major]

4 tips on every scramble.

Goodbye sub 4 averages.

 

[Kirjava]

This really has nothing to do with that thread.

It's pretty much the same issue.

It's strictly about the pyraminx and how we see it as a puzzle.

Yeah, it was a pretty silly reason to filter pyraminx scrambles.

 

[Stefan]

I'm aware that this line of reasoning also means that the name of the "Magic" and "Master Magic" events should probably likewise be changed to "Modified Magic" and "Modified Master magic".

I don't see why. We're doing *the* official task as stated in the subtitles ("link the rings" and "unlink the rings").

 

[cmhardw]

I don't see why. We're doing *the* official task as stated in the subtitles ("link the rings" and "unlink the rings").

I meant that the line of reasoning I proposed considers the event in terms of what portion of the legal scrambled states of the puzzle are allowable for the scramble of event itself. I guess this does not apply at all to magics since the event of magic does not even have scrambles.

 

[Radu]

To all: Don't forget to read my last statement. I don't pretend the regulations to be changed or to introduce a new event (with all scrambled tips). I'm not trying to make a revolution or something totally impractical.

I just want to have a debate if the way we use to scramble the pyraminx is wrong since we started? Because this is what I think. We let the scrambling program choose if a tip will be scrambled or not. We don't just scramble it and say "oops, that's an easy scramble"...the scrambling program just skips to scramble it!

To put it another way. We get a scramble...there are 2 options it was generated:

1. The actual way the generator gives us a scramble now: U R B' L U u l where the scramble total skips to scramble a tip (which I consider a puzzle itself)...actually 2 in this case.

OR

2. We can have the same scramble: U R B' L U u l, which actually comes from U R B' L U u b3 l r3 which obviously is the same. But there is a big difference in the way it gets scrambled. Every tip get scrambled, but the b3 and r3 obviously are not displayed as they bring the puzzle in a solved state.

I think the first option is the one we use now. Or if the generator works under the 2nd principle, then just let me know and close the thread . It's a big difference in the way those two work. This is why I said in the first post "unless we consider a solved puzzle a scrambled one, if the scramble gives us a solved puzzle".

So, I agree with this scramble if it comes from the 2nd way of scrambling, not from the first one. And why do I say that is best explained in the example in the first post with the 100 tips puzzle.

 

[Kirjava]

The actual way the generator gives us a scramble now: U R B' L U u l where the scramble total skips to scramble a tip (which I consider a puzzle itself)...actually 2 in this case.

 var tips=["l","r","b","u"];
  for (i=0;i<4;i++) {
   var j = Math.floor(Math.random() * 3);
   if (j < 2) {
    scramblestring[n] += tips[i] + ["","'"][j] + " ";
    picmove(4+i,1+j,n);
   }
  }

Looks like they all get scrambled to me.

 

[Stefan]

U R B' L U u l, which actually comes from U R B' L U u b2 l r2 which obviously is the same.

Not at all obvious to me, can you explain how that is the same and what you mean with "comes from"?

 

[vcuber13]

i think hes saying he doesnt understand why there is a 1/3 chance that a tip will be solved

 

[Radu]

Not at all obvious to me, can you explain how that is the same and what you mean with "comes from"?

Sorry Stefan, you're right...I intended to say b3 and r3. I will just edit.

@kirjava - that code is interesting and should pretty much clear up everything. I'm not that good of a programmer, but does it clearly result everything gets scrambled or he randomly chooses between all 4 tips? So there can be 4 scrambled or none.?

 

[vcuber13]

i believe it generates a random number (1-3) for each of the tips and for example for the u tip, 1 is u 2 is u' and 3 is nothing.

 

[Kirjava]

@kirjava - that code is interesting and should pretty much clear up everything. I'm not that good of a programmer, but does it clearly result everything gets scrambled or he randomly chooses between all 4 tips? So there can be 4 scrambled or none.?

Everything is scrambled. This entire thread is based upon an assumption you made that was wrong.

 

[Brunito]

Tim i dont think you have to say byebye to sub4avgs cause i think Odder Oka and Me and maybe some other pyrmainxer can do sub4 avg 100 with only 4tip it is not that hard i think

 

[Carrot]

Tim i dont think you have to say byebye to sub4avgs cause i think Odder Oka and Me and maybe some other pyrmainxer can do sub4 avg 100 with only 4tip it is not that hard i think

I guess I can just edit the scrambler and then do an avg100 and see what happens ;D