×How can I use the roach to redirect Halph sufficiently?

×All of your assumptions are true: You need to use the roach to make Halph take the long way back to the starting plate.

This room showcases a quirk in Halph’s path-finding: When returning to his “home tile” after opening a door (or being interrupted from doing so), he will look for the closest path to any tile adjacent to his home tile, instead of looking for the closest path to the home tile itself. This is similar to how he behaves when issued a “follow me” command by Beethro.

Another thing about Halph’s path-finding is that he will only recalculate the path to his destination when something blocks his current path. (The only exception to this rule is when he is following Beethro, then he will recalculate his path every turn, but that doesn’t apply here).

×I already thought of that solution. But... The three tiles that you want to block don't touch. So I have available: Beethro's body (check), the roach (check). And now my sword can't reach the third hot tile

× Solution (I think): three sword can block of a tile, even if it can't hover above the tile itself.

×First let me repeat what Dying Flutchman just said: The sword can block off a hot tile even if it isn’t placed on a hot tile.