[Daily puzzle] Sierpiński carpet
912 sats \ 3 comments \ @south_korea_ln 23h science
Today, another problem from Math Horizons.
The following figure illustrates a Sierpiński carpet, a fractal structure. We start with a square of side length equal to 1 unit length. At each iteration, one cuts a square into 9 subsquares and deletes the middle square. In the limit, this will be a Sierpiński carpet.
Two questions:
  • How many squares remain at each iteration?
  • An ant, located at one corner of the carpet wants to travel as fast as possible to the diagonally opposite corner without falling into a hole. She can travel in any direction for that. She can do better than a distance of two unit lengths (simply following the outer edges). What is the shortest distance she can cover?
Previous iteration: #751726 (the answer I had was indeed by letting the bulb become warm, but I appreciate the other even more creative solutions to think out of the box)
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320 sats \ 0 replies \ @Scroogey 20h
The shortest path is
If you look at that path in iterations 1-3, you see that it cuts through solid squares in two symmetric ways, in such a way that the next iteration's hole just barely misses it. Due to the fractal nature, this repeats with all subsequent iterations, infinitely.
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301 sats \ 0 replies \ @didiplaywell 20h
1 - Taking as "squares" only the individual ones and not the composed ones, and understanding that the white ones are not counted:
2 - Due to how the fractal iterates, the shortest path for the ant is always the same: going straight to any corner of the central square that from his POV is placed transversally, and then to the opposite corner.
For example, if the ant starts in the lower left corner, it must go straight to the lower right corner of the central square, then to the opposite corner. That path is safe for all iterations of the fractal, and always the shortest.
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0 sats \ 0 replies \ @Aardvark 23h
8/64/512?
I have no idea about the ant, so probably 42.
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