Quote:
Originally Posted by Mathgod
Maybe I misunderstood the question. When you say "it is impossible to add another quarter to the table without requiring an overlap" do you mean coins can be slid on the table and shifted around, as long as they never overlap? Or are they all locked into a fixed position?
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This is a good question, imo.
I misread it initially, and my solution assumed the coins were touching each other, which wasn't stated in the problem.
In many ways this reminds me of a reservoir engineering problem. The spaces in between the coins would be porosity (just 2d instead of 3d). Having the coins not touching would make it a non-physical solution, but you'd have perfectly sorted matrix with exactly equal grain sizes.
I think the answer probably flows from your observation about the distance between the coins. I don't see the solution but have the following observations.
The number of coins given is 400, or 4:1 for the number of initial coins. That ratio isn't physical, but 4 is 2 squared, and we know the distance between the centers of the coins can't be bigger than 2D, because then you could fit another coin.
I think the proof will involve squaring that distance and demonstrating that 4x is enough coins to cover it somehow.
I didn't spoiler this because its just middle of the night musings, and could be completely unhelpful to anyone...