Re **Puzzle 25** I tried to answer what to me was a slightly more plausible formulation which reversed a cat heating the room to "the bed is hot enough for the cat?" (maybe looking for it). This isn't very good as it suggests the answer is T|F. But I don't think this necessarily excludes a map so I got \\(f_\ast\\) as the range of such temps. I need a better formulation.
And thanks to all for the unbeatable, brain-mangling fun! :).