I'm teaching an elementary proofs course, and would love an excuse to show the recursive construction of a space-filling curve:
I'm having trouble coming up with a question to which this a good answer. My first thought was just to ask for a curve that hits every cell, enters at the bottom left and exits at the top left, but unfortunately such curves are two-a-penny. So my question:
Is there a natural and comprehensible question to which a recursive construction like the one I've shown is the answer (or failing that a reasonably natural answer)?
The kind of question I'm looking for is something like:
Consider a $2^n\times 2^n$ grid of squares. Prove that there is a curve consisting of straight lines joining the centres of squares with ???? property going from the bottom left to the top left of the grid
This is not a specific question, but an important concept that it can be an example for. You have to be careful commuting limits and functions. The limit of the areas of the curves in the series is zero because every curve in the series has zero area, but the area of the limit curve is $1$.
With the edit, how about prove there is a path with no straight segment longer than $2$ squares?. When you increase $n$ by $1$, you replace the path inside the square with a zigzag one through the new squares. Unfortunately, when you do the subdivision you don't enter and leave at the same point as you did before, so it takes a little more work to show the recursive path works.