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48
votes
4answers
4k views
Why is a circle in a plane surrounded by 6 other circles
When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number?
I'm ...
5
votes
3answers
180 views
Number of point subsets that can be covered by a disk
Given $n$ distinct points in the (real) plane, how many distinct non-empty subsets of these points can be covered by some (closed) disk?
I conjecture that if no three points are collinear and no four ...
25
votes
1answer
836 views
Dividing a square into equal-area rectangles
How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$?
The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
2
votes
1answer
194 views
Which internal angles can a lattice polygon have?
I am wondering if for a lattice polygon an internal angle can take any value? If no which ones not and why?
I guess there will be some restrictions due to the discrete nature of the grid but I am ...
2
votes
1answer
142 views
Brouwer's fixed point theorem implies Sperner's lemma
I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. The proof should be understandable by an undergraduate.
Thanks in advance!
0
votes
1answer
25 views
Number of faces of $n$ congruent disks
If I have $n$ disks, all of the same radius, how many faces (i.e. maximally connected regions) can the induced arrangement have? For example for 3 disks, it could have 7 bounded faces, but what is ...
