Advanced EOLine - Part 1

by Richard Taylor : 2022-08-12

In the previous article I shared some tools for helping ZZ solvers move towards optimal EOLine solutions. Once you have mastered that, what else is there to learn?

Green-Blue and Red-Orange Neutrality

Unlike CFOP, where many solvers learn to make the cross on any of the 6 faces (colour neutrality), the prospect of making a line on any face and orienting all 12 edges seems very, very difficult.

But it seems intuitive that if the line on white looks tricky with green or blue on front, then it might be easier to make a line at 90 degrees to that instead (line on white but edges oriented with red or orange on front).

So let's look at some numbers. I did a billion random scrambles and counted the "bad" edges on the z-axis (green-blue) and also on the x-axis (red-orange). I found that the proportion of scrambles where the x-axis count was less than the z-axis count was 0.329 and where it was less or equal was 0.671.

Just in terms of the number of bad edges then, you have a 32.9% chance of doing better by switching from Z to X, and a 67.1% chance of not doing worse.

Since you can do "better" if you are able to solve on either green-blue or red-orange 32.9% of the time, that means the chance of doing better on at least one solve in a round of 5 is 86.4%, which is huge. So it definitely seems worth the trouble. Obviously the number of bad edges alone doesn't paint the whole picture, but quantifying the "niceness" of the line edges will have to wait for further analysis.

If we are convinced that there is value in learning to solve EOLine on two different axes, let's look at some more detail. On those billion scrambles these are the proportions of "bad" edges on the x-axis for each of the possible counts on the z-axis.

Z edges  X0      X2      X4      X6      X8      X10     X12
0        0.0019  0.0650  0.3389  0.4531  0.1414  0.0000  0.0000
2        0.0010  0.0524  0.3115  0.4508  0.1714  0.0129  0.0000
4        0.0007  0.0415  0.2782  0.4507  0.2057  0.0229  0.0003
6        0.0005  0.0322  0.2415  0.4517  0.2414  0.0322  0.0005
8        0.0003  0.0229  0.2057  0.4507  0.2783  0.0415  0.0007
10       0.0000  0.0129  0.1716  0.4507  0.3114  0.0525  0.0010
12       0.0000  0.0000  0.1405  0.4527  0.3396  0.0652  0.0020

Clearly you are unlikely to get fewer than 4 bad edges on red-orange if there are 12 on blue-green, but also you are unlikely to get more than 8. Here are the chances of doing the same or better given a particular "bad" edge count on one axis.

Z    Better or same
0    0.0019
2    0.0534
4    0.3204
6    0.7259
8    0.9578
10   0.9990
12   1

And of doing strictly better.

Z    Better
0    0
2    0.0010
4    0.0422
6    0.2741
8    0.6795
10   0.9466
12   0.9980

So it looks like you are unlikely to do better on the second axis if there are 6 or fewer "bad" edges on the first axis. But you are also unlikely to do worse if there are 6, so it might be worth considering if the line edges are awkward. Also on the positive side, if there are 10 or more "bad" edges on the first axis then you are almost certain to do better on the second axis.

Is it only slightly better though? Swapping a 12 for a 10 isn't a particularly fantastic trade. Here are the chances of doing better by 4 and 6 edges, as well as just the minimum of 2. (actually better by 2 or more, 4 or more, 6 or more)

Z    Better 2+    Better 4+    Better 6+
0    0            0            0
2    0.0010       0            0
4    0.0422       0.0007       0
6    0.2741       0.0327       0.0005
8    0.6795       0.2288       0.0231
10   0.9466       0.6352       0.1845
12   0.9980       0.9327       0.5931

So it is good news. There is a 59.3% chance that the second axis will have 6 or fewer "bad" edges if the first axis has 12. And a 63.5% chance of 6 or fewer if the first has 10. Those are significant chances of improvement.


Always solving EOLine with white on bottom and green or blue on front will get you a long way. Most of the time you will get 4, 6 or 8 edge cases and you know how to plan those effectively. But if you get unlucky and get a couple of 10 edge cases then it can kill your average.

Being able to solve EOLine with red or orange on front as well could save the day in this situation. There is almost a 95% chance that red-orange will have 8 or fewer "bad" edges if blue-green has 10; and more than 63% chance that there will be 6 or fewer.

"There's no such thing as a free lunch" they say. And the cost is that learning a second axis is hard. As if learning one wasn't hard enough. I am just starting this process... some days it goes well, some days it seems impossible. I'll write a follow up if I make it out the other side!


I got so focussed on reducing the number of bad edges that I didn't show the overall proportion of each number of bad edges when you only look at blue-green versus looking at the best of blue-green and red-orange. Here is that data:

Bad Edges    % blue-green    % best axis
0             0.05            0.10
2             3.22            6.27
4            24.17           39.57
6            45.12           45.12
8            24.17            8.77
10            3.22            0.18
12            0.05            0.00

Note that the proportion of 6-bad cases is the same but there is a significant shift from 8 and 10 to 4 and 2.

Update 2023

After trying this for a good 3 months I was making progress, but just suddenly gave up. I guess progress was too slow. But the main reason was that every solve just felt hard. I was making a significant number of blunders still - mixing up good/bad edges, forming pairs wrongly, putting pairs in the wrong place - and rarely seemed to get into a flow like I had before.

I don't know if I just baked in one axis too much before starting this or if it's really a lot harder than learning blue-green neutrality (which I did a long time ago and fairly early into learning ZZ). Anyway, I failed, but I gave it a good try. Within a month or two of reverting back to 1-axis ZZ my PB had improved by about 2 seconds on what it was before. So, no harm done.