https://i.imgur.com/yE43nun.png
https://i.imgur.com/NTfbt5U.png
Note how the middle of the donut aligns with the bottom or top of the cup.
> don't see some obvious proof of why your approach is correct
I’m not a mathematician; maybe I got lucky with my approach. Crucially this is a brain teaser so it plays by the rules of a game. I formulated an hypothesis by starting from the assumption that they give you a satisfactory fighting chance.
I think it's easy to see that if the central hole of the doughnut were either smaller or larger, or if the handle of the cup was smaller or larger, these splits would not work.
> I’m not a mathematician; maybe I got lucky with my approach. Crucially this is a brain teaser so it plays by the rules of a game. I formulated an hypothesis by starting from the assumption that they give you a satisfactory fighting chance.
Sure, that's fair, it's just that your original explanation for how the solution is to swap the problem from splitting the cup to splitting the doughnut was so intriguing and puzzling that got me curious.