It occurs to me that the question about whether non-trivial cycles exist for the collatz conjecture can be restated as these two questions (details on how this relates to the collatz conjecture can be found here): Is there a general method for determining how many distinct values of $t_1, t_2, \...

Consider this function: $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}$$ Here is a StackExchange post with more information about ...