I wanted to ask more questions about variations on This Problem. If you hadn't seen that one, check it out. It's required reading (and probably more fun).
In the original problem, these were the rules about sign placement:
- Each sign placed must point to the next sign placed
- Placed signs become obstacles, so future signs cannot point through them
- Every new sign turns right (90 deg clockwise of last sign)
- An older sign may not become blocked by newer signs
In the original problem, Weather Vane found that the maximum density of signs on a 9x9 board (starting in the middle) was:
50 / 81
So first an easy question:
A) If the board were instead very large (many billions of cells, for example), what limit could we place on the maximum sign density?
Let us say the limit has to be a simple fraction. Same is true for the second question:
B) Again on a very large board, what limit could we place on the sign density if we eliminate the 4th rule and allow older signs to be blocked?
Hint/suggestion on part B:
In part A, the answer comes from knowing that there's a cute little pattern that fills a corridor 3 cells wide. For a large board, you can imagine that pattern zig-zagging back and forth and filling any space. Around the perimeter, there may some little inefficiencies from turning around, but for a very large board that has no meaningful impact on the overall density. That same idea applies for part B. Imagine something like a 3-cell wide perimeter where you can connect adjacent patterns as needed, and then only really worry about making a pattern that fits next to itself well. As for the start location, don't worry too much about starting in the middle.
The reason I thought this makes for a interesting question is that without the 4th rule, you wouldn't think it would be hard to do better than part A, but something about it makes that very difficult. It took me days to finally come up with a repeating pattern that was more dense, and it was sort of an "ah-ha moment" which is why I thought it might make for a good question. Also it is very different from working in a small 9x9 board.
To help answer part B you may (or may not) want to try solving the original 9x9 board without the 4th rule. I'm not asking that as it's own question, but if you want to try it I guess we can call that question C.
If you want to post some examples, I found that using Excel's "conditional formatting" helps for readability, now that things can become more confusing. Like this: