Btw is the footnote a joke? I don't really get it:
The sum of the digits 1 to 9 is 45[1]
[1] This is a secret that Simon only tells his closest friends.
https://gazj.substack.com/p/python-and-the-legend-of-zelda?s...
Article doesn't contain a mathematical proof (only a brute force one), but I wrote one up. Spoilers: https://news.ycombinator.com/item?id=30639211
I guess the Zelda puzzle is different because in Königsberg you can revisit islands, just not recross bridges. But that feels like something you can finesse somehow. . . . Ah, just swap nodes & edges, right? Squares : bridges :: sides : islands. Indeed, that lines up not just the restriction but the goal too.
EDIT: Oh your second link is a much nicer solution. But still it feels like there is a relationship to Königsberg.
Swapping nodes and edges doesn't work, because many of the Zelda nodes have 3 or 4 edges, but an edge is defined as 2 endpoints. It doesn't make sense to talk about an edge with 3 or 4 ends.
The proof of non-solution to the Zelda puzzle is a simple checkerboard argument. The room's dimensions are 13 x 9, both odd, so all the corners are the same color (call it black) and there is one more black square than white. And the prize square replaces a white square. So there are two more black squares than white squares, making the problem unsolvable, since you must always alternate visiting white and black squares. The statues are a red herring - there are two on each color and so they don't affect this proof.
A tricker version asks what square remains (unique up to symmetry) when covering a chessboard with 21 trominoes, each of which covers 3 adjacent board squares, i.e. 1x3 or 3x1.