A thought experiment targeted at the cubing community

[bcube]

Hi, imagine if you will:

there is a new event introduced in the track & field: Random Sprint. It takes the sprint events - 100 m, 200 m, 400 m - and these distances are randomly selected by a computer for each competitor in the race.

In this experiment, the race consists of:

• Lane 1: 100 m
• Lane 2: 200 m
• Lane 3: 400 m
• Lane 4: 100 m
• Lane 5: 200 m
• Lane 6: 400 m

There are no competitors in lane 7 and 8, so 6 competitors in total are competing in the Random Sprint event.

The race is on. Competitors in lanes 1, 3 and 5 clock the time of 25 seconds. Competitors in lanes 2, 4 and 6 clock the time of 60 seconds.

Rankings for the Random Sprint event are following:

• shared first place for competitors in lanes 1, 3 and 5 (result: 25 seconds)
• shared fourth place for competitors in lanes 2, 4 and 6 (result: 60 seconds)

Do the rankings seem right to you? Vote in the poll above.

 

[Clock_Enthusiast]

Is this an analogy for different optimal move counts between FMC competitions or scrambles with easier solutions between groups held at the same (or different) competitions?

 

[bcube]

scrambles with easier solutions

What's your analogy for scrambles with easier solutions in the Random Sprint event?

If your argument is: "competitor in lane 1 runs a shorter distance than competitor in lane 2", then consider a modified thought experiment with no rankings whatsoever (the rest of the race is the same as described in post no. 1 above), as rankings are not relevant at all for the process of determining what distance will be run by what competitor in Random Sprint.

 

[bcube]

Thanks to all voters for voting, much appreciated. As of now, there are 3 Yes votes and 7 No votes, with 90% of votes made till day 1 of the poll, so I am going to analyze below what the results indicate to me.

The poll was made to verify/deny my hypotheses and to try to understand better how the cubing community think.

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Question: how does track & field race relate to cubing?
Answer: the thought experiment is supposed to be the first approximation of the FMC ranking system.

Q: why do you use an approximation and not a full analogy?
A: cube theory is a little bit more complex to simulate an analogy of the FMC ranking system through the track & field race. However, the core principles of these two areas are the same.

Q: what do you mean by cube theory?
A: both speedcubers and FMC solvers can be experts in cube-theory-for-specific-methods. This is not what I mean. By cube theory I mean a theory not specified to certain methods, a computer science related cube theory if you will.

Hypothesis 1: there is no correct answer to the poll without stating the context first.

As you can see, the context is not stated in the original post. From a speedcuber perspective, I can clearly understand how the Yes answer is correct. Therefore, I can understand the logic behind 3 voters voting Yes.

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Hypothesis 2: the FMC ranking system has been introduced by speedcubers (FMC solvers not considering cube theory well enough would fit too).

The FMC ranking system has been introduced in 2004 and hasn´t changed since then. I don´t blame the 2004 authors for it, as I think neither Ron, nor Tyson have been thinking through the computer science consequences their FMC ranking system is causing because having the same rules for speedsolving and FMC events makes perfect sense from speedsolver's point of view. I would love to hear from @Ron if my hypothesis is correct and what he thinks about the FMC ranking system from today´s perspective.

Q: but the 3x3x3 God´s number in HTM has been known only since 2010. Does it mean that the results from 2004 to 2010 would not be fair/correct anyway?
A: see post no. 15 from the cube theory link above. You don´t need to know the God´s number for it, you need an optimal solver for it. And optimal solvers have been available since the late 1990´s.

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Hypothesis 3: the WCA will fail in its vision to become a recognized sport organization under the current circumstances.

The WCA is actively trying to become a recognized sport association. If/when the WCA reaches that status in the future, I can't wait to see some FMC competitor suing the WCA in front of an independent (i.e. not speedcubing-related) jury, because in my ipinion, the jury has no other choice than claiming the WCA being discriminatory (with whatever consequences a verdict might require (such as getting back its "sport status" until the fairness issues are resolved)). This opinion is based on my interpretation of poll results, specifically interpreting that 7 out of 10 "people in independent jury" would vote for the FMC ranking system being disciminatory.

 

[bcube]

Q: hold on a second. You're saying that there is nothing wrong with the Random Sprint race?
A: that's correct.

Q: but how is the race fair if a competitor in lane 1 runs a shorter distance than a competitor in lane 2?
A: the problem is not in the race itself, it lies in the process of determining who was the winner of the race, who finished second, third and so on.

Q: so who was the winner of the race, according to you?
A: first we must divide the race into 3 separate races. Competitor in lane 1 is the winner in the 100 m race, competitor in lane 5 is the winner in the 200 m race, and competitor in lane 3 is the winner in the 400 m race. It doesn't matter that all these competitors were racing at the same time in one race.

 

[Mike Hughey]

But you said "who was the winner of the race". The WCA agrees with you on who the winner of the race is - there is no competition where people get different scrambles and are ranked against each other for the competition's result. The place the WCA disagrees with you is on overall ranking. And the argument there is the same as for any other event. Since scrambles are random, some are easier than others, so ranking can't reasonably take that into account - to do so would unnecessarily complicate the ranking charts. 2x2x2 is certainly every bit as affected by randomness for overall ranking as FMC. If you care that much about overall ranking, the answer is simple: go to more competitions and you will at least be able to get close to the top of the rankings if you're good enough.

 

[bcube]

But you said "who was the winner of the race". The WCA agrees with you on who the winner of the race is - there is no competition where people get different scrambles and are ranked against each other for the competition's result.

In the thought experiment, you can see that competitors in different lanes get different distances. But I know what you meant, and I agree with you that if competitors in all lanes get the same distance, the winner of the race can be easily determined - it would be the first competitor at the finish line.

If you care that much about overall ranking, the answer is simple: go to more competitions and you will at least be able to get close to the top of the rankings if you're good enough.

Consider a modified thought experiment: the competitor in lane 3 finishes in 25.01 seconds instead of 25 seconds flat, the rest of the experiment is the same as in the original post above. This will be the first but also the last race for the competitor in lane 3 in his athlete career. Don´t you think the competitor in lane 3 deserves to be declared a new world record holder in the 400 m dash, instead of not even winning the Random Sprint race despite running 4 times faster than the winner of the race in lane 1? Equally importantly, if you allow competitors in lanes 1 and 5 to be ranked above the competitor in lane 3, you are giving me (as a ranking viewer) a wrong impression that performance-wise, competitors in lanes 1 and 5 are better - how am I supposed to know the competitor in lane 3 run 4 times faster than a competitor in lane 1 if I only see the competitor´s time in rankings (which are called "Random Sprint rankings")?

Since scrambles are random, some are easier than others, so ranking can't reasonably take that into account - to do so would unnecessarily complicate the ranking charts.

Do the 100 m dash rankings, 200 m dash rankings and 400 m dash rankings seem unnecessarily complicated to you (if not, what is your concern exactly)? Also, these are not giving me a wrong impression of what runner is better.

2x2x2 is certainly every bit as affected by randomness for overall ranking as FMC.

As stated in post no. 4, the thought experiment is supposed to be the first approximation of the FMC ranking system, not a ranking system for a speedsolving event.

 

[Mike Hughey]

It may be your intention that it apply only to FMC, but in all honesty it applies to all events. Some might be affected more than others; for instance, 2x2x2 is almost certainly more affected than FMC. But I suppose FMC would require fewer categories, if you go by optimal move count, so you would only have to publish a few lists.
But honestly even that is not fair. There are easy to find 18-movers and hard to find 18-movers. It's like asking if you can even rank 400 meter races against each other when done at different altitudes. Someone with a lucky 18 move scramble and lesser skill might easily beat someone with much greater skill but an unlucky 18 move scramble. Categorizing by optimal move count is likely not even the best way to categorize difficulty.

 

[bcube]

To keep us on track, I will only quote these:

There are easy to find 18-movers and hard to find 18-movers. [...] Someone with a lucky 18 move scramble and lesser skill might easily beat someone with much greater skill but an unlucky 18 move scramble.

I don´t see what seems to be the problem with that.

Categorizing by optimal move count is likely not even the best way to categorize difficulty.

If you think my goal is to categorize difficulty then I am afraid you missed my point completely.

 

[bcube]

Q: Mike is right with his 2x2x2 example. What´s your reason to differentiate between FMC and speedcubing events?
A: great question! While FMC can be measured and quantified, speedcubing events can be only measured.

Q: what do you mean by quantification, though?
A: answering "how good the achieved result is" on that specific scramble considering all possible outcomes. As I hope you can imagine, the discussion about what are all possible outcomes for a specific scramble in speedcubing events can get very complex very fast. On contrary, in FMC it is relatively simple.

Q: why?
A: because in FMC, we already know what the best possible outcome on that specific scramble is - therefore we can answer the "how good the achieved result is" question by comparing the best possible outcome with the achieved competitor´s result. In speedcubing events, the best possible outcome is uknown, we only know competitor´s result on that specific scramble.

 

[Billabob]

This again!!

 

[xyzzy]

Man, I actually started writing a wall of text effortpost before the poll closed (edit: not "closed" per se, but since the motivation for setting up the poll was revealed) but I sat on it instead of finishing it… Maybe another time lol.

 

[bcube]

before the poll closed

I am probably misunderstanding what you meant by that, but the poll in this thread isn't supposed to be time-limited. If it seems closed to you, feel free to make it open again as a moderator (it seems to be open for me as a forum member).

2x2x2 is almost certainly more affected than FMC. But I suppose FMC would require fewer categories, if you go by optimal move count, so you would only have to publish a few lists.

Statistically speaking, this can be easily calculated. Without looking at the data, I would guess all 8 distance distribution values (from 4 to 11) for 2x2x2 already occured during the WCA competitions, and it's likely there will be more distance distribution values for FMC in the future. I can't specify the time interval for the "future" given the amount of variance associated with random sampling. However, it should take about 14.1 million FMC scrambles on average to get 9 distance distribution values (from 11 to 19).

 

[bcube]

Without looking at the data, I would guess all 8 distance distribution values (from 4 to 11) for 2x2x2 already occured during the WCA competitions, and it's likely there will be more distance distribution values for FMC in the future. I can't specify the time interval for the "future" given the amount of variance associated with random sampling. However, it should take about 14.1 million FMC scrambles on average to get 9 distance distribution values (from 11 to 19).

To give us some perspective what it means: I was told there are currently a little bit over 8500 FMC scrambles in the WCA database (which was established in 2014 for FMC scrambles according to this post). Even at the pace of 10000 FMC scrambles per year - and we are nowhere near that pace right now - it would take next 1409 years to get the 14.1 millionth FMC scramble.

I was also told that during those FMC 8500 scrambles in the WCA database, there was no occurence of optimal 14-mover. In the table below you can see it was expected to get 1 such scramble (because 8500 > 6188), so now you know what the phrase "given the amount of variance associated with random sampling" means.

Distance	Expected Probability [%]	Expected Odds [1 in X]
18	67.0489 %	1.5
17	27.7443 %	3.6
19	3.4680 %	28.8
16	2.5432 %	39.3
15	0.2112 %	473.3
14	0.0161 %	6188
13	0.0012 %	81353
12	0.00009 %	1071272

What you can also read from the table is that more than 99 % of FMC scrambles will be either 18-mover, 17-mover, 19-mover or 16-mover. In the next table you can see some stats for the mean of 3 format, and you can compare some stats with the previous table for the single attempt format. Row in red is the highest number of moves known to me (e.g. ColombiaNationals2014), row in green is the lowest number of moves known to me (e.g. DanishFMChampionship2022). I don´t have an access to the WCA database. Data in the first table are taken from cube20.org.

Combination	Expected Probability [%]	Expected Odds [1 in X]	Number of Permutations	Example Order	Number of Moves
2×18 + 1×17	37.4178 %	2.7	3	18, 18, 17	53
3×18	30.1422 %	3.3	1	18, 18, 18	54
1×18 + 2×17	15.4832 %	6.5	3	18, 17, 17	52
2x18 + 1x19	4.6772 %	21.4	3	18, 18, 19	55
1x18 + 1x17 + 1x19	3.8708 %	25.8	6	18, 17, 19	54
2x18 + 1x16	3.4299 %	29.2	3	18, 18, 16	52
1x18 + 1x17 + 1x16	2.8386 %	35.2	6	18, 17, 16	51
3x17	2.1356 %	46.8	1	17, 17, 17	51
2x17 + 1x19	0.8008 %	124.9	3	17, 17, 19	53
2x17 + 1x16	0.5873 %	170.3	3	17, 17, 16	50
1x18 + 1x16 + 1x19	0.3548 %	281.9	6	18, 16, 19	53
1x18 + 2x19	0.2419 %	413.4	3	18, 19, 19	56
1x17 + 1x16 + 1x19	0.1468 %	681.2	6	17, 16, 19	52
2x16 + 1x18	0.1301 %	768.6	3	16, 16, 18	50
1x17 + 2x19	0.1001 %	999.0	3	17, 19, 19	55
2x16 + 1x17	0.0538 %	1858	3	16, 16, 17	49
1x16 + 2x19	0.0092 %	10869	3	16, 19, 19	54
2x16 + 1x19	0.0067 %	14925	3	16, 16, 19	51
3x19	0.0042 %	23809	1	19, 19, 19	57
3x16	0.0016 %	62500	1	16, 16, 16	48

 

[bcube]

Q: I am a current FMC competitor. How would your FMC ranking system affect me?
A: no change in a single attempt only format. As for more-than-a-single-attempt format, I can think of 2 paths to consider:

1. restrict competitors from knowing them the optimal move count until all solves in a round are finished
2. tell comptetitors the optimal move count for a round, starting from attempt number 2

Option 1 works well in theory but is difficult to follow in practise, so I personally would go with option no. 2. Since all competitors don´t know the optimal move count for attempt 1, and all competitors know the optimal move count for the rest of their attempts in a round, no one is being discriminated.

Q: why would you tell competitors the optimal move count in option no. 2?
A: because I want to prevent the situation in which some competitors know the optimal move count and some competitors don´t.

Q: I used to be an FMC competitor. How would your FMC ranking system affect me?
A: as long as your scrambled cube state is known, you are eligible to be ranked in a single attempt format. For instance, I was able to find all cube states for 29 to 20 single format results from this page. As for a mean of 3 format, if you are lucky to have the same optimal move count in all 3 scrambles per round (it is expected to be more than 30 % of a time as stated in the second table in the post above) then you are eligible to be ranked in a mean of 3 format on the WCA ranking page. For the cases not covered above, I can think of 2 paths to consider:

1. aside from catogories like 18, 17, 19, 16 and so on, create a category "N/A" and place the results there
2. don´t create a non-integer category name and let the results to be seen in your WCA competitor page, similarly to how e.g. Old MBLD results are treated

To keep FMC rankings fair while keeping them as simple as possible for ranking viewers/non-cubers on the WCA rankings page, I would personally go with option no. 2. That way, the ranking viewers/non-cubers would only see catogories like 18, 17, 19, 16 and so on for both the single attempt format and the mean of 3 format on the WCA rankings page.