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25 sats \ 3 replies \ @south_korea_ln OP 9 Oct 2024
Visual representation of the problem.
The 5 points were chosen at random here.
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100 sats \ 2 replies \ @Undisciplined 9 Oct 2024
I don't understand why it isn't just the square root of two. Can't the points be arbitrarily close to the corners?
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73 sats \ 1 reply \ @south_korea_ln OP 9 Oct 2024
In that case, where would you put the 5th point?
Let me rephrase the problem, just in case:
The closest two points of five in a square cannot be further than X apart.
Find X.
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0 sats \ 0 replies \ @Undisciplined 9 Oct 2024
That clarification helps. I'm still working on my coffee this morning.
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276 sats \ 5 replies \ @Undisciplined 9 Oct 2024
I'll take half of the square root of 2. One point arbitrarily close to each corner and one in the center.
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269 sats \ 1 reply \ @SimpleStacker 9 Oct 2024
I'll second this, with the added note that if the square is a closed set you don't have to say arbitrarily close to the corner :)
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20 sats \ 0 replies \ @Undisciplined 9 Oct 2024
Indeed. I took "inside" as implying an open set.
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239 sats \ 2 replies \ @Scroogey 9 Oct 2024
This assumes that the optimal placement is the four corners and the center.
The 'pigeonhole principle' can be used to prove your answer is correct no matter the placement:
Divide the square into four equal smaller squares (in the obvious way).
At least one of these squares must contain two points.
The maximum distance of two points in a square of length 1/2 is 1/sqrt(2).
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37 sats \ 0 replies \ @south_korea_ln OP 9 Oct 2024
Was hoping for someone to bring up this pigeonhole principle.
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0 sats \ 0 replies \ @Undisciplined 9 Oct 2024
I like it. I'm quite out of practice at doing proofs. Maybe these daily puzzles will get me back to peak form.
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