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?
