#proofing with pigeonhole principle
royal mangoBOT
median cairn
<@&286206848099549185>
wary plank
@median cairn Sometimes, the problem gives you the bare minimum. Why 10 points in particular? Can you place 9 points into a square such that no two points are closer than 0.48 units?
I will say, splitting the square into smaller squares is the correct approach.
median cairn
hmm im not sure if 9 would work because the sidelengths would be too small for it to be 0.48 units apart
since itd all be 0.11 i think
per smaller square
the proof i wrote right now splits the square into 4 smaller squares of 0.5 length but then uses the general pigeonhole principle, since there will eventually be 2 or more points in the same space within 0.50, so those two are within 0.48 then
if theyre within 0.50
wary plank
if theyre within 0.50
That would not necessarily follow. If you have 2 points in a square of side length 0.5, then simply place them at two corners and they are 0.5 apart from each other.
wary plank
hmm im not sure if 9 would work because the sidelengths would be too small for i...
You want the side lengths to be too small, since you are trying to prove two points are closer than 0.48 units from each other.
I don't think splitting into four squares will work. If you have 2 vertices in a square of side length 0.5, you cannot conclude that those points are less than 0.48 units apart. No can you do anything if you have 4 points in a square of side length 0.5.
The general pigeon-hole principle does say that 3 points will be in the same square of side length 0.5, but I don't think you can do anything from that.
Here, splitting the square into 4 squares does not help at all.
Primarily because you are forced to focus on pairs of distant points, or a trio of distant points.
And you cannot see the cluster of 4 points obviously close to each other.
wary plank
hmm im not sure if 9 would work because the sidelengths would be too small for i...
(9 will work)
median cairn
If i do 9 wont each side have to be 0.333?
For it to all add up to 1? Rhat was my main concern
wary plank
Yes. Each square will have side length 1/3.
Now, what does the pigeon-hole principle tell you?
@median cairn