What is the routine for calculating the result when the
fraction is less than or equal to a half ?

What is the routine when the fraction is greater than a
half?

Can you express that in algebra?

If $x$ is the fraction value what is the result in terms of
$x$ for the two cases above?

For each denominator (bottom number of the fraction) which
fractions are equivalent to other fractions with a smaller
denominator?

Which fractions form a loop?

Which fractions do not belong to the loop but track into
it?