I'm very bullish on Google long-term, but the other frontier labs' approach of (successfully!) solving a raft of prestigious mathematics problems is top-notch strategy IMHO. With almost anything else, they would never break through the current of "Pfft, that's not real intelligence!" goalpost-shifting that's out there. It takes a level of demonstrated ability that cannot be plausibly denied to make that kind of impact. Unfortunately, causing some of the brightest minds in the world to worry about whether their profession has a future may have been necessary to achieve that level (I say nothing about whether this is ethically acceptable).

I would be extremely impressed by a robot that swam the Atlantic or climbed Everest, as would almost any human alive. If you have a robot that can do one of those things, you should absolutely do it as a publicity stunt.