In building my table of 4s, I used some hand-searching, and some computer tables. The tables created combinations of various 1- and 2-four combinations, computing the values of each combo. I'm not going to share all of that code here (though it worked pretty well). Instead, here is the list of 'particles' that I used as input.
Manipulation of these particles generates lots of good solutions. A purely machine-generated solution to the standard problem is probably not possible, as there are an infinite number of ways to manipulate each 4.
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