Not only does the problem become easier, using the right notation enables us to do things that we just couldn't do before. Humans (or at least me) can only hold

Not only does the problem become easier, using the right notation enables us to do things that we just couldn't do before. Humans (or at least me) can only hold a few simple things in our head at once. If the problem is too complicated, we can't deal with it, and the only way to approach it is to split it down into simpler abstractions that we can hold in our head all at once. I think this is a large part of what mathematics is - taking concrete concepts that are hard to hold in our head, and turning them into an abstraction that describes their important and general properties in a way that is simple enough that we can keep them in our head. So much of mathematics is the pursuit of different abstractions and general properties of things. I can juggle with 3 balls, and I'm able to do a passable Mills-Mess, but I'm not sure I have the experience to judge the utility of your juggling notation. However, it certainly seems to be doing a similar thing. Your notation contains exactly the same information as the entire pattern but it can actually be kept in ones head and worked with. Even being able to juggle Mills-mess, I'm still not really sure what's going on, and certainly struggle to imagine the whole pattern. I'm not surprised it took you 2 and half pages to describe!