Combining opposite color solving with extended cross solving
About the extended cross | Examples | Solving the extended cross | Opposite color solving
What is opposite color solving?
This is an idea that was proposed to me by Richard Patterson, that I now swear by. I always use this strategy in my solving, and have even used it in official competitions. My fastest official competition solve using my opposite color is 12.87 seconds achieved at the Rubik's World Championships in 2005. Basically all opposite color solving means is that you solve either your regular cross/2x2x2 color or also its opposite.
Why use opposite color solving?
The below function is exactly why you want to use opposite color solving.
Graph obtained from this website
This function is F(x) = 1 - (1 - x)^2 and is exactly the reason why you want to use this strategy. The variable x is your chance (expressed as a number from 0 to 1) of getting anything on one layer. It can be the chance you get a good cross, or a good 2x2x2. It can be the odds of you solving an extended cross, whatever you want to look at.
Notice that using two faces does not double your chances for seeing something, but rather increases it as an inverted quadratic (a learning curve basically).
For example, I have a roughly 25% chance of doing an extended cross if I restrict myself to using only one face. Plug this as 0.25=x into the formula and you get F(x) = 7/16 = 0.4375. This means that if I solve extended cross with a 1/4 frequency, then by choosing to solve the opposite face too I now have a 7/16 = 43.75% chance of seeing an extended cross on either my regular color or its opposite.
How to get used to opposite color solving
The hardest thing about opposite color solving is that the colors on F,R,B,L will be seemingly in the "wrong" order. This can be confusing at first, but over time you get used to it. I would not let this initial difficulty deter you though, as your odds of getting even just a good cross will increase by the graph shown above. Say you have a 10% chance of getting a really good, super fast cross on your solve based on your solving style/method/etc.. Well if you choose to also solve your opposite color you now have a 19% chance of getting a super fast cross. And you did this by doing nothing different other than getting used to a different order in the side colors of your F2L on some of your solves. This is such a small change to drastically improve your percentage for all kinds of good cases. The math behind this technique, and the odds of seeing good cases for such a small change, is exactly why I now use this method all the time.
Using extended cross and opposite color solving together
The purpose of this page is to convince you that solving just the extended cross is not enough. Call this advanced extended cross solving if you want, but I think it should be absolutely required to solve opposite color as well, if you plan on solving with extended cross. As a combination this strategy is so powerful, and I see no reason to only use one part of it and neglect the other part. I now advocate using both strategies together, as it really can aid in your solving times.