# Why can't we detect infinite loops in the busy beaver problem?

I was reading about the busy beaver problem the other day and I'm confused as to why we can't keep an array of Turing machine states that the machine has been through and simply iterate through it at each new machine state to see if it exactly matches a state we've previously seen. That ought to tell us there's an infinite loop, no? Then following this, shouldn't we be able to solve the problem for any given Turing machine, instruction set and tape?

• its easier for some to understand when realizing that "solving" the busy beaver problem would allow solving the halting problem
– vzn
Sep 5 '14 at 22:48
• For a lot of algorithms it's easy enough, but some will be utterly pathological on every level. Like not knowing if there chaotic, or provably unprovable, or they're testing the coldatz conjecture. "{1}, {2, 1}, {3, 10, 5, 16, 8, 4, 2, 1}, {4, 2, 1}" now what? Either "{…, 425…682, 331…520, …} - False" or "{5, 16, 8, 4, 2, 1}, {6, 3, 10, 5, 16, 8, 4, 2, 1}, …" - True. Nov 23 '18 at 19:07