# Busy beaver construction system [closed]

I want to construct a busy beaver TM which has the alphabet $\Sigma = \{1,2, X\}$ and $X$ is the blank symbol. I want to do this using 3, 4 and 5 states (means one just for halting, so 3 states = 2 'normal', 1 for halt). I am looking for kind of a system (theoretical pattern) to do this...is there anything like that?

I managed to construct busy beavers but I did not manage to come up with something like a system.

• Software recommendation questions are offtopic here. – Raphael Apr 10 '16 at 17:31
• I'm not looking for software! I am looking for a certain pattern I can reuse everytime to construct TMs... – 今天春天 Apr 10 '16 at 17:32
• There isn't one. – Raphael Apr 11 '16 at 8:04
• Why was this closed? They are looking for a theoretical pattern not software recommendation. I agree that there is no theoretical pattern, but that sounds like the appropriate "answer" to this CS question. – sligocki Apr 11 '16 at 13:17
• Answer: It is impossible for there to be any computable pattern to finding Busy Beavers. If the Church-Turing thesis is true, that means there is guaranteed to be no pattern a human could find either. Each Busy Beaver example is a unique snowflake. – sligocki Apr 11 '16 at 13:21