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So the riddle i was asked is: you have a circle, with a radius of 9, prove that you cant pack 101 points inside of it, without having at least a pair of dots with a distance less than 2 between them.

to be clear, i know the true solution, so thats not the question.

while breaking the head i came up with the following: you put a square around said circle, making its side 18, ill treat it as 9x9 because we need the dots to be 2 units apart so the relations between stays the same from 18 - 2 and 9 - 1

here is the part that we have a disagreement on:
i pack said cube with 1x1 cubes, ensuring all the space is used and 1 unit apart, and get a total of 100, which should mean that you cant go over that in said circle which is smaller.

the mathhead that gave me the riddle says that he is unsure that square packing is optimal, and maybe if he fills it with hexagons he might get more than 100 points, making this way invalid.

I dont have the mathematical knowledge to prove or disprove that, so here i am with this question. Does the way work? or is he right?

Reference picture for my idea

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    $\begingroup$ This isn’t very clear, unfortunately... but in your tiling of the $18 \times 18$ square with $81$ $2 \times 2$ squares, please note that two elements in the same square need not be at distance less than two (consider the opposite corners). $\endgroup$
    – Aphelli
    Commented May 27, 2021 at 7:29
  • $\begingroup$ well each dot on the 2x2 square is exactly a distance apart of 2, and so the diagonal will always be more than 2. $\endgroup$
    – KlesierTheSurvivor
    Commented May 27, 2021 at 7:35
  • $\begingroup$ To prove the statement using the pigeonhole principle, you want to split the disc into 100 (or fewer) regions. That way you know that with 101 points, there must be some region with at least two points in it. For this to work as a proof of the statement, you need those regions to have diameter less than 2, so that any region with 2 (or more) points forces two points to be less than 2 apart. As Mindlack points out, your 81 squares have diameter of 2.82..., so do nothing to prove the statement. $\endgroup$
    – Jaap Scherphuis
    Commented May 27, 2021 at 7:55
  • $\begingroup$ Certainly, there are denser ways to pack circles. What you are lacking is a proof that the denser packings also don't fit 101 circles in the big circle. $\endgroup$
    – Eric Towers
    Commented May 29, 2021 at 0:01

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