# Big cube BLD - solved centers analysis



## cmhardw (Apr 12, 2021)

In this post I will give the results of an analysis I ran using R to answer a couple unanswered questions for 4x4x4 BLD and 5x5x5 BLD.

Background
On 4x4x4 BLD it is common practice at the beginning of the solve to rotate the cube such as to maximize the number of solved centers when viewing the cube in your reference orientation. Before 2010, and as I remember it, Mike Hughey popularized doing the same thing on 5x5x5 BLD. At the start of a 5x5x5 BLD solve, you rotate the cube into one of the twelve orientations that will not result in void cube parity such as to maximize the number of solved centers when viewing the cube in your reference orientation. You will then permute the central-most centers as pieces and will need to memorize their unsolved state during the memorization phase.

4x4x4 Analysis - Purpose
Let's start with an analysis of the 4x4x4. When scrambling the cube and holding the orientation fixed, each center has a 1/6 probability to land on its home face. Multiply this by 24 centers and the expected number of solved centers is 4. If instead you rotate the cube such as to maximize the number of solved centers, how many centers are expected to be solved? Furthermore, what is the distribution of number of solved centers from 0 centers solved through 24 centers solved?

4x4x4 Analysis - Methodology
I simulated 100 million (10^8) 4x4x4 random states and counted the number of solved centers in all 24 possible orientations. I then compared the maximum number of solved centers to the reference or starting orientation to build a comparison of the two BLD methods.

4x4x4 Analysis - Results

Below is a graph showing the distributions of solved centers for both fixed orientation as well as rotating to maximize solved centers.



Here are some statistics for the fixed and solved center approaches.

Fixed: mean=3.999978 centers solved; sample standard deviation=1.865174 centers
Rotating for max solved centers: mean=7.919506 centers solved; sample standard deviation=1.169982 centers

I note that the distributions do not appear to be normal, and in that case a sample standard deviation is not very meaningful. I would be happy to take requests from others on how best to analyze the variance of this data. I am also happy to give my .RData file to anyone who would like to analyze this themselves and report back here in this thread. For a full distribution table from this analysis please see my my google sheet.

This builds a strong case for why it is good technique in 4x4x4 BLD to rotate the cube to maximize the number of solved centers. Notice that rotating will give you on average about 4 more solved centers than using fixed orientation, and it also lowers the standard deviation in number of solved centers. The conclusion is that you more reliably have more solved centers when you rotate for maximum solved centers rather than used a fixed orientation.

5x5x5 Analysis - Purpose
Recently I participated in Abhijeet Ghodgaonkar's 5x5x5 BLD video where multiple solvers showed their methods on the same scramble. After the video was posted I was contacted by multiple cubers who asked about the fact that I had rotated the cube at the start of the solve to maximize solved centers. I think this technique is not well known, but was popularized by Mike Hughey sometime before 2010. I have been using this as my main 5x5x5 BLD starting approach since that time. I intuitively knew that this very often gave me more solved centers than fixed orientation, but wanted to quantify the difference. This analysis does just that.

5x5x5 Analysis - Methodology
I simulated 100 million (10^8) 5x5x5 random states and counted the number of solved centers in all 12 orientations where the permutation parity of the corners matches the permutation parity of the midges. I then compared the maximum number of solved centers to the reference or starting orientation to build a comparison of the two BLD methods.

5x5x5 Analysis - Results

Below is a graph showing the distributions of solved centers for both fixed orientation as well as rotating to maximize solved centers.



Here are some statistics for the fixed and solved center approaches.

Fixed: mean=7.999607 centers solved; sample standard deviation=2.637653 centers
Rotating for max solved centers: mean=12.67163 centers solved; sample standard deviation=1.575379 centers

I note that the distributions do not appear to be normal, and in that case a sample standard deviation is not very meaningful. I would be happy to take requests from others on how best to analyze the variance of this data. I am also happy to give my .RData file to anyone who would like to analyze this themselves and report back here in this thread. For a full distribution table from this analysis please see my google sheet.

This quantitatively backs up my intuition from using this method that rotating the 5x5x5 at the start of a BLD solve to maximize solved centers, choosing from only the 12 orientations that do not result in a void cube parity, will give you more solved centers on average than using fixed orientation. Rotating for maximum solved centers you can expect 4.67 more solved centers on average, and a lowered volatility in number of solved centers. Note on my distribution table that the minimum number of solved centers when rotating is 8, and when using fixed orientation having fewer than 8 solved centers represents 44.3% of all scrambles.

Conclusions
I propose that 5x5x5 Blindfolded solvers strongly consider rotating the cube at the start of the scramble to maximize for solved centers. Doing so will give you on average 4.67 more solved centers than if you used fixed orientation. Rotating will also lower the volatility in number of solved centers, and thus add predictability in the number of center targets required in your memorization.


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## abunickabhi (Apr 12, 2021)

Wow good insights Chris. I hope floating orientation becomes a norm for odd layer NxN cubes as well (well, odd layer cubes bigger than the 3x3).

Generally in my 5BLD solves, I do +center execution first (exec order is +xcmw), so it will be good if the [MES] gen algs cancel a few moves with the first uf+ center commutator (although I cannot expect this to happen every solve, as the MES interchange move of the +center may not cancel with the orientation fixing alg).

The current WR of 5BLD had lot of solved centers from the base orientation, so its a lucky solve.


Spoiler: Scramble 5BLD WR



L U2 Rw F' Bw' Lw2 Rw' Dw2 Uw2 L2 Dw2 L' Uw' U' Dw2 Bw2 Dw' F Rw L2 U Dw2 Fw' Uw2 Fw Rw2 Dw F R2 F2 B Rw Bw2 F2 Lw F2 Dw2 Lw2 Uw2 B2 Lw' F Uw' F' Lw F2 R' L B' Bw Lw U2 Dw R Lw' Fw D Rw2 B' 3Rw' 3Uw (13 solved center pieces)


but the current AsR of 5BLD has 0 xcenters solves, so switch to one of the 12 orientations would have resulted in better solve than 2:39.12, although Kaijun exec is one of the fastest in the world (memo is the fastest too)


Spoiler: Scramble 5BLD AsR



R' F2 U Rw U2 Uw' Bw' Dw2 Lw Rw' Uw Lw Uw2 Bw' B' Lw' B Fw' Lw2 B F' Lw2 F U' Rw' Bw Dw2 D2 F2 Dw' R2 Rw Dw F2 L B2 Lw' R' Rw' L' Uw' Fw2 Rw2 L F Rw2 Lw' Bw2 Fw' Dw2 Uw L' Bw2 D Bw' B2 Lw2 Rw B' Uw 3Rw 3Uw'(5 solved center pieces, none of them are xcenter type)


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## Mike Hughey (Apr 12, 2021)

Wow, Chris, nice to see these results!

How hard would it be to compare the option of fully reorienting 5x5x5 BLD, for all possible orientations? I know there were at least a few people who were actually going that far - doing the parity fix as part of the solution. (I think I remember that, in particular, François Courtès did this.) This obviously adds some to the solve portion, so it's a more substantial tradeoff, but I am curious how much help it actually provides with regard to solved centers as compared to just the 12 non-parity orientations. I always assumed it would slow me down too much to justify my doing it, but I also figured that maybe better solvers might be able to get enough out of it to justify the extra effort.


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## cmhardw (Apr 12, 2021)

Mike Hughey said:


> How hard would it be to compare the option of fully reorienting 5x5x5 BLD, for all possible orientations?



This wouldn’t be too difficult of a code change, but it would probably take me about a week to run the simulations and tabulate the results. I’ll definitely look into this! I also am almost certain I can reuse and repurpose my 4x4 simulations for this, but I want to examine this more thoroughly before trying that route.


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## cmhardw (Apr 15, 2021)

Here is a histogram showing the differences between the Maximum number of solved centers and the reference orientation. This effectively shows you for each of the 100 million simulated solves what the difference is between the number of solved centers when choosing the optimal orientation and using fixed orientation every solve.

I also added the distribution table to my google sheet

Note that the expected value for the difference between fixed orientation and optimal orientation is 4.67 centers, but there is quite a lot of tail weight to the right. With 19.4% probability you will have a difference of 7 or more solved centers. With 11.4% probability you will have a difference of 8 or more solved centers in the optimal orientation vs fixed orientation.

I think this method will be very similar to Xcross in that you will not use it every solve, but on some solves it will be a mistake not to take advantage of a large difference in solved centers in the optimal orientation compared to your fixed orientation.


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## Kit Clement (Apr 15, 2021)

Just wanted to add here that standard deviation is fine to report for any kind of non-normal data, it's just that the "empirical rule" interpretation (68-95-99.7%) is of course not a valid interpretation. It still holds weight as a measure of variability for any sample of data and still gives a rough idea of the typical distance from the mean. Variance is better if you want something surely unbiased, I suppose, but I prefer the interpretability of a standard deviation and the bias is almost always negligible. Great analysis, and I'm glad you did the paired difference analysis too, that was my first thought reading through the original post as well!

A practical question for you (and any other 5BLDers reading this), what do you think is the number of additional solved centers necessary for the additional memorization of the center swap and execution to break even?


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