Is it possible to raise the complexity of what you can mentally process? I know that I can solve some simple systems of linear equations in my head by visualizing the terms and moving them. But if I don't concentrate enough, it gets blurry quite fast.
In the same way, I can remember some structure of an algorithm, a mathematical proof or a speech, but not with details. In the same way, I tend to forget the first elements if I add new ones.

I feel that, by saying that, I mix at least two things:


what I have heard as a "mind limit", around 3 or 4 elements anyone can process at the sale time, or recognize visually without counting them
memory, which you can theoretically train to store large amounts of elements


What can be trained, and how, to be able to process more elements at the same time? Even if there's a hard mind limit, is it possible to make playing mentally with N elements much clearer, and how?
I think the answer you are looking for is chunking.


  A chunk is defined as a familiar collection of more elementary units
  that have been inter-associated and stored in memory repeatedly and
  act as a coherent, integrated group when retrieved (Tulving &amp; Craik,
  2000). It is believed that individuals create higher order cognitive
  representations of the items on the list that are more easily
  remembered as a group than as individual items themselves.


The repeated exposure unconsiously creates chunks of many elements that are stored in long term memory, which are retreived as whole units in the working memory. In this case the limit is 3-4 chunks, each composed of many elements.

It is sometimes possible though, if needed, to consiously decompose the chunk into its elements in working memory, but in this case only 3-4 items of the decomposed chunk can be active in the working memory.