Is there a simple algorithm to decide which of the numbers
$$a \uparrow ^b c \text{ and } d \uparrow ^e f$$
is the bigger one ?
Using the hyperoperation, the numbers can be denoted with
$$H_{b+2}(a,c)\text{ and } H_{e+2}(d,f)$$
I tried using the recursive definition of $H$
$$H_n(a,b) = H_{n-1}(a,H_n(a,b-1))$$
and induction to get useful properties, but without substantial success.
If the given numbers are very large, the following heuristic should give the correct result in many cases :
If $b>e$, then the first number is bigger. If $b=e$ and $c>f$, then the first number is bigger. If $b=e$ and $c=f$, it is trivial to compare the numbers.
Of course, this heuristic cannot hold in all cases.
Any ideas?