This problem has been plaguing me right before I go to sleep for a while now, please help!!

My question is...

"Suppose we take 10 points inside a unit'square. Show that some two of the points are at distance at most sqrt(2)/3 from each other."

I am satisfied that the answer can be obtained using the pigeonhole theorem, so that by spliting the unit square into 9 boxes the greatest distance between vertices of adjacent boxes is sqrt(2)/3 but that suggests to me that there should be a configuration of points such that this is true.

It is this configuration that I am looking for!!

Extending the problem, how can I work the problem out for n points?

For example, I know (hope) the answer for n=3 is 2*sqrt(2-sqrt(3)).

For 4 points the answer is simply 1 (a point at each vertex).

etc.

Can I use the pigeonhole argument to show the answer for everycase (I've only met it today so I'm not sure really how to use it). How can I find the configuration of points such that my answer is correct.

I hope I have provided a coherent question for you. Any resources or answers are greatly appreciated!

Yours, sleepless.

Nope. This is true for any configuration of points. This is what the problem is about.

First of all get yourself acquainted with the Pigeonhole principle. It's at

https://www.cut-the-knot.org/do_you_know/pigeon.shtml

Second, note that if 10 points are placed into 9 boxes, at least one box contains at least 2 points. And what was the size of the small box you said?