Here are some examples of the intuitive insights I mentioned when talking about Beeping Booping Busy Beavers:

Fermat’s Last Theorem (as a conjecture) is an example of a level 1 question, answerable by an ordinary Busy Beaver

The Twin Primes Conjecture is an example of a level 2 question, which requires a level 2 Busy Beaver which has access to a level 1 Busy Beaver oracle which answers questions about whether a level 1 beaver halts. The Beeping Busy Beaver equivalence clarifies this. Level 2 questions are generally more difficult than level 1 questions.

An example of a level 3 question is ‘Does there exist a C such that 3^A-2^B=C has an infinite number of solutions in the integers?’ The equivalence to Beeping Booping Beavers shows the level. Level 3 questions are generally more difficult than level 2 questions, but they’re thought about much less often in mathematics. I came up with this possibly original example because no simple famous one comes to mind.

Unrelated to that I previously made a cool animation illustrating a possibly better fill pattern along with code with generates it. There’s now a new plug-in system for Cura which should make it straightforward to add this pattern. If anyone implements that it would be much appreciated. People have criticized this pattern because it isn’t all curves like gyroid, and I initially assumed myself that should be changed, but now I’m not so sure. Gyroid isn’t really the mathematical gyroid because it gets very thin in places because of how layers work which gives it points of weakness, and the vibration of having those sharp corners is much less of an issue with input shaping and other 3d printer improvements. What matters most now is strength of support for the amount of support material used and this pattern should score very well on that benchmark.