Not sure if this is a joke, but actually that is guaranteed to be true. It is proven that for all n: BB(n+1) >= BB(n) + 3. But it is not proven that BB(n+1) >= BB(n) + 4, haha.
The point stands: the hard part is proving that all the programs with longer runtime than your upper bound will never terminate, and once you've solved that, getting the exact value is just a little extra work
For arbitrary n, that proof is arbitrarily hard, even undecidable for large enough n. Again though, for the specific case n=6, that difficulty has not yet been demonstrated, especially if you're willing to accept probabilistic arguments instead of rigorous proofs. n-by-n checkers is PSPACE-complete but the specific case n=8 that people actually play, has been completely solved using computers.
If you want an unproven-but-almost-certainly-correct upper bound on BB(6), consider BB(12).
Not sure if this is a joke, but actually that is guaranteed to be true. It is proven that for all n: BB(n+1) >= BB(n) + 3. But it is not proven that BB(n+1) >= BB(n) + 4, haha.
The point stands: the hard part is proving that all the programs with longer runtime than your upper bound will never terminate, and once you've solved that, getting the exact value is just a little extra work
For arbitrary n, that proof is arbitrarily hard, even undecidable for large enough n. Again though, for the specific case n=6, that difficulty has not yet been demonstrated, especially if you're willing to accept probabilistic arguments instead of rigorous proofs. n-by-n checkers is PSPACE-complete but the specific case n=8 that people actually play, has been completely solved using computers.